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

<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font><br/>  for <font color="Olive" title="b2">L</font> being   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b3">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>)</span>)</span><br/>  for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>  st  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b1">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b4">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Olive" title="b5">g</font>, <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>) holds <br/><font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">g</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b1">T</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b2">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>)</span>)</span><br/>  for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>  st  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Olive" title="b2">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b3">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Olive" title="b4">g</font>, <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>) holds <br/><font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Olive" title="b2">P</font>,<font color="Maroon" title="c2">T</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">L</font> be   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b2">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Olive" title="b3">g</font>, <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>) holds <br/><font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">T</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Olive" title="b2">g</font>, <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>) holds <br/><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c6">g</font>, <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>) implies <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c7">R</font> =  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>);<br/>
<span class="kw">assume </span><a NAME="E1:86"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c6">g</font>, <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)</span><br/>

<a NAME="E2:86"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T50" target="_self" title="POLYRED:th.50">Th50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:86"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">h</font>
 <span class="kw">and </span><br/><a NAME="E5:86"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c8">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D7" title="REWRITE1:def.7">REWRITE1:def 7</a></span>;<br/>
<a NAME="E6:86"/>
<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">h</font>,<font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="ref" href="rewrite1.html#T24" title="REWRITE1:th.24">REWRITE1:24</a></span>;<br/>
<a NAME="E7:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A3</font></span>: 
<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">h</font>,<font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T22" title="REWRITE1:th.22">REWRITE1:22</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E8:86"/>
<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c8">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="rewrite1.html#T22" title="REWRITE1:th.22">REWRITE1:22</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E9:86"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:86"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="ref" href="rewrite1.html#T16" title="REWRITE1:th.16">REWRITE1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:86"/>
<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D4" title="REWRITE1:def.4">REWRITE1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
