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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font>, <font color="Olive" title="b3">b3</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b1</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Olive" title="b2">b2</font> holds <br/><font color="Olive" title="b1">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive" title="b3">b3</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Olive" title="b2">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive" title="b3">b3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</font>, <font color="Maroon" title="c4">b3</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font> implies <font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:4"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:4_1"/>
( <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><span class="kw">suppose </span></a><a NAME="E1:4_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:4_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="4_1_2"><span class="kw">suppose </span></a><a NAME="E1:4_1_2"/>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span><br/><div class="add"><a NAME="E2:4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c3">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="pre_poly.html#D10" title="PRE_POLY:def.10">PRE_POLY:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">k</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E4:4_1_2"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">and </span><br/><a NAME="E5:4_1_2"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> holds <br/><font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D9" title="PRE_POLY:def.9">PRE_POLY:def 9</a></span>;<br/><div><span class="lab"><font color="Green" title="E6">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">l</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:4_1_2_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4_1_2"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:4_1_2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:4_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>;<br/><a NAME="E8:4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4_1_2"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E9:4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_1_2"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="pre_poly.html#D9" title="PRE_POLY:def.9">PRE_POLY:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E10:4_1_2"/>
<font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c4">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D10" title="PRE_POLY:def.10">PRE_POLY:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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