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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="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#V2" title="BAGORDER:attr.2">admissible</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#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</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">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="b2">L</font><br/>  for <font color="Olive" title="b5">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>  st  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Olive" title="b5">P</font>,<font color="Olive" title="b1">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</font> holds <br/>( <font color="Olive" title="b4">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">T</font> &amp; ( <font color="Olive" title="b4">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <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>) or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Olive" title="b4">g</font>,<font color="Olive" title="b1">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">T</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#V2" title="BAGORDER:attr.2">admissible</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#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</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">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="Olive" title="b1">L</font><br/>  for <font color="Olive" title="b4">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>  st  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Olive" title="b4">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>,<font color="Olive" title="b3">g</font> holds <br/>( <font color="Olive" title="b3">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Olive" title="b2">f</font>,<font color="Maroon" title="c2">T</font> &amp; ( <font color="Olive" title="b3">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <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>) or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Olive" title="b3">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c2">T</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#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</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">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><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="Maroon" title="c3">L</font>)</span>)</span>  st  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Olive" title="b3">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>,<font color="Olive" title="b2">g</font> holds <br/>( <font color="Olive" title="b2">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">T</font> &amp; ( <font color="Olive" title="b2">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <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>) or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">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">  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>  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="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> holds <br/>( <font color="Maroon" title="c5">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font> &amp; ( <font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <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>) or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">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"> (  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> implies ( <font color="Maroon" title="c5">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font> &amp; ( <font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <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>) or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</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="c6">P</font>,<font color="Maroon" title="c2">T</font>);<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">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="c6">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>,<font color="Olive" title="b2">g</font> holds <br/> for <font color="Olive" title="b3">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>)  st <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Olive" title="b2">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">T</font>;<br/>
<span class="kw">assume </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</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">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font> &amp; ( <font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <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>) or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>)<span class="kw"> such that </span><br/><a NAME="E3:78"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
( <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E62">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:78_1"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">thus </span><a NAME="E2:78_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:78_1_1"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">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="c6">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font> implies  for <font color="Olive" title="b1">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>)  st <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Maroon" title="c11">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c2">T</font> )</span><br/><span class="kw">assume </span><a NAME="E1:78_1_1_1"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</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">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>)  st <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Maroon" title="c11">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c2">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">p</font> be    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</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="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon" title="c11">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c2">T</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:78_1_1_1"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">and </span><br/><a NAME="E3:78_1_1_1"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">and </span><br/><a NAME="E4:78_1_1_1"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c11">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c2">T</font></span><br/><a NAME="E5:78_1_1_1"/><span class="lab"><font color="Green" title="E68">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A10</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">q</font> = <font color="Maroon" title="c12">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">q</font> = <font color="Maroon" title="c12">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T15" title="FINSEQ_1:th.15">FINSEQ_1:15</a></span>;<br/><a NAME="E8:78_1_1_1"/><span class="lab"><font color="Green" title="E70">A11</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E9:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A12</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E10:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A13</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">h</font> = <font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q</font></span>)</span>;<br/><a NAME="E11:78_1_1_1"/><span class="lab"><font color="Green" title="E73">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E8:78_1_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E12:78_1_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E5:78_1_1_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="txt" href="#E8:78_1_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E13:78_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78_1_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>;<br/><a NAME="E14:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_1_1_1"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E4:78_1_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E11:78_1_1_1"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="txt" href="#E10:78_1_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">h9</font>, <font color="Maroon" title="c17">g9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:78_1_1_1"/><span class="lab"><font color="Green" title="E75">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h9</font>,<font color="Maroon" title="c17">g9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:78_1_1_1"/><span class="lab"><font color="Green" title="E76">A17</font></span>: 
<font color="Maroon" title="c16">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <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>
 <span class="kw">and </span><br/><a NAME="E18:78_1_1_1"/>
<font color="Maroon" title="c17">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T2" title="RELSET_1:th.2">RELSET_1:2</a></span>;<br/><span class="lab"><font color="Green" title="E77">A18</font></span>: <font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h9</font>,<font color="Maroon" title="c17">g9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:78_1_1_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c16">h9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
;<br/><div><span class="lab"><font color="Green" title="E78">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:78_1_1_1_2"/><span class="lab"><font color="Green" title="E79">A20</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q</font>
 <span class="kw">and </span><br/><a NAME="E2:78_1_1_1_2"/><span class="lab"><font color="Green" title="E80">A21</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>)</span><br/><a NAME="E3:78_1_1_1_2"/>
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:78_1_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E2:78_1_1_1_2"><span class="lab"><font color="Green" title="E80">A21</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:78_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A22</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> <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="#E8:78_1_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:78_1_1_1_2"/>
<font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:78_1_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E1:78_1_1_1_2"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:78_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A23</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> <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="#E8:78_1_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:78_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:78_1_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E2:78_1_1_1_2"><span class="lab"><font color="Green" title="E80">A21</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E8:78_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A24</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:78_1_1_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="txt" href="#E4:78_1_1_1_2"><span class="lab"><font color="Green" title="E81">A22</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:78_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:78_1_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E1:78_1_1_1_2"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E10:78_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:78_1_1_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="txt" href="#E6:78_1_1_1_2"><span class="lab"><font color="Green" title="E82">A23</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E11:78_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A25</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_1_1_1_2"><span class="lab"><font color="Green" title="E83">A24</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>;<br/><a NAME="E12:78_1_1_1_2"/>
<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_1_2"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E13:78_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1_1_1_2"><span class="lab"><font color="Green" title="E80">A21</font></span></a>, <a class="txt" href="#E11:78_1_1_1_2"><span class="lab"><font color="Green" title="E84">A25</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E21:78_1_1_1"/>
 <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>;<br/><a NAME="E22:78_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c16">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><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>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:78_1_1_1"><span class="lab"><font color="Green" title="E76">A17</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E23:78_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">h9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q</font></span>)</span> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:78_1_1_1"><span class="lab"><font color="Green" title="E76">A17</font></span></a>, <a class="txt" href="#E19:78_1_1_1"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a>, <a class="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">q</font> = <font color="Maroon" title="c14">q</font> as    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E11:78_1_1_1"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="txt" href="#E20:78_1_1_1"><span class="lab"><font color="Green" title="E78">A19</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>;<br/><a NAME="E26:78_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E9:78_1_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E27:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A26</font></span>: 
<font color="Maroon" title="c19">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E28:78_1_1_1"/><span class="kw">then </span>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c18">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></span>;<br/><a NAME="E29:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A27</font></span>: 
<font color="Maroon" title="c18">h</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E4:78_1_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E8:78_1_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E27:78_1_1_1"><span class="lab"><font color="Green" title="E79">A26</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E30:78_1_1_1"/>
<font color="Maroon" title="c18">h</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:78_1_1_1"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="ref" href="polyred.html#D13" target="_self" title="POLYRED:def.13">Def13</a></span>;<br/><a NAME="E31:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A28</font></span>: 
 ex <font color="Olive" title="b1">r</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 <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c18">h</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c11">g</font>,<font color="Olive" title="b1">r</font>,<font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D7" target="_self" title="POLYRED:def.7">Def7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">h</font> = <font color="Maroon" title="c18">h</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <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> ;<br/><a NAME="E33:78_1_1_1"/>
<font color="Maroon" title="c11">g</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c20">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:78_1_1_1"><span class="lab"><font color="Green" title="E81">A28</font></span></a>, <a class="ref" href="polyred.html#T43" target="_self" title="POLYRED:th.43">Th43</a></span>;<br/><a NAME="E34:78_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c20">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E35:78_1_1_1"/>
<font color="Maroon" title="c11">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:78_1_1_1"><span class="lab"><font color="Green" title="E80">A27</font></span></a>, <a class="ref" href="polyred.html#T27" target="_self" title="POLYRED:th.27">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:78_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E5:78"/><span class="lab"><font color="Green" title="E63">A29</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T25" target="_self" title="POLYRED:th.25">Th25</a></span>;<br/>
<a NAME="E6:78"/><span class="lab"><font color="Green" title="E64">A30</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E63">A29</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c9">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:78"/><span class="lab"><font color="Green" title="E65">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 ;<br/>
<a NAME="E9:78"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E65">A31</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:78"/><span class="lab"><font color="Green" title="E66">A32</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E64">A30</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E65">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <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>) or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:78_2"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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">  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font></span><br/><a NAME="E2:78_2"/><span class="kw">then </span>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/><a NAME="E3:78_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A33</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><span class="kw">assume </span><a NAME="E4:78_2"/><span class="lab"><font color="Green" title="E68">A34</font></span>: 
 not  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:78_2_1_1"/>
(  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:78_2_1_1_1"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:78_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_2"><span class="lab"><font color="Green" title="E68">A34</font></span></a>, <a class="ref" href="termord.html#T6" title="TERMORD:th.6">TERMORD:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:78_2_1_1_2"/><span class="lab"><font color="Green" title="E69">A35</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:78_2_1_1_2"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T6" title="TERMORD:th.6">TERMORD:6</a></span>;<br/><a NAME="E3:78_2_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D2" title="TERMORD:def.2">TERMORD:def 2</a></span>;<br/><a NAME="E4:78_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A36</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><a NAME="E5:78_2_1_1_2"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T6" title="TERMORD:th.6">TERMORD:6</a></span>;<br/><a NAME="E6:78_2_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D2" title="TERMORD:def.2">TERMORD:def 2</a></span>;<br/><a NAME="E7:78_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">T</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">T</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a>, <a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></span>;<br/><a NAME="E8:78_2_1_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_2_1_1_2"><span class="lab"><font color="Green" title="E69">A35</font></span></a>, <a class="txt" href="#E4:78_2_1_1_2"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></span>;<br/><a NAME="E9:78_2_1_1_2"/><span class="kw">then </span>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_2"><span class="lab"><font color="Green" title="E68">A34</font></span></a>, <a class="ref" href="termord.html#D2" title="TERMORD:def.2">TERMORD:def 2</a></span>;<br/><a NAME="E10:78_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A37</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_2_1_1_2"><span class="lab"><font color="Green" title="E69">A35</font></span></a>, <a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></span>;<br/><a NAME="E11:78_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="polyred.html#E48"><span class="lab"><font color="Green" title="E42">Lm15</font></span></a></span>;<br/><a NAME="E12:78_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></span>;<br/><a NAME="E13:78_2_1_1_2"/><span class="kw">then </span>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E66">A32</font></span></a>, <a class="ref" href="polyred.html#T26" target="_self" title="POLYRED:th.26">Th26</a></span>;<br/><a NAME="E14:78_2_1_1_2"/><span class="kw">then </span>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_2"><span class="lab"><font color="Green" title="E67">A33</font></span></a>, <a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E15:78_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:78_2_1_1_2"><span class="lab"><font color="Green" title="E71">A37</font></span></a>, <a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:78_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:78"/>
( <font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <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>) or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c2">T</font>) <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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