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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">L</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><br/>  for <font color="Olive" title="b4">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  not <font color="Olive" title="b4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">b1</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b3">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b4">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b5">u9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">u</font> holds <br/><font color="Olive" title="b3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b5">u9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b5">u9</font> ) holds <br/> for <font color="Olive" title="b5">x</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/>  for <font color="Olive" title="b6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">L</font>  st <font color="Olive" title="b6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">u</font></span>)</span>) holds <br/> <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Olive" title="b3">b2</font>,<font color="Olive" title="b5">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b5">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><br/>  for <font color="Olive" title="b3">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  not <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">b1</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b4">u9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">u</font> holds <br/><font color="Olive" title="b2">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">u9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">u9</font> ) holds <br/> for <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">u</font></span>)</span>) holds <br/> <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Olive" title="b2">b2</font>,<font color="Olive" title="b4">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b4">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  not <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b2">u9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">u</font> holds <br/><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">u9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">u9</font> ) holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">u</font></span>)</span>) holds <br/> <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Olive" title="b2">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Olive" title="b2">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b1">u9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u</font> holds <br/><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">u9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">u9</font> ) implies  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">u</font></span>)</span>) holds <br/> <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Olive" title="b1">x</font>)</span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font>
 <span class="kw">and </span><br/><a NAME="E2:29"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E3:29"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 for <font color="Olive" title="b1">u9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u</font> holds <br/><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">u9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">u9</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">x</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">u</font></span>)</span>) holds <br/> <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Olive" title="b1">x</font>)</span>)</span></span><br/>

<a NAME="E4:29"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<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/>
<a NAME="E5:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">sb2</font> =  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">sb1</font> =  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font></span>)</span>);<br/>
<span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">u</font></span>)</span>) holds <br/> <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">x</font>)</span>)</span></span><br/>

<a NAME="E6:29"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">u</font></span>)</span>) implies  <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">x</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E7:29"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">u</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="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">x</font>)</span>)</span></span><br/>

<a NAME="E8:29"/>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#T13" target="_self" title="POLYNOM2:th.13">Th15</a></span>;<br/>
<a NAME="E9:29"/><span class="kw">then </span>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">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="E11:29"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:29"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E13:29"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">u</font> = <font color="Maroon" title="c5">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>;<br/>
<a NAME="E15:29"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E16:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E17:29"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E15:29"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E18:29"/><span class="kw">then </span>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">k9</font> = <font color="Maroon" title="c10">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<a NAME="E20:29"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c12">k9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
;<br/>
<a NAME="E21:29"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#T13" target="_self" title="POLYNOM2:th.13">Th15</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_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="E1:29_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">n</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_1"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">x</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:29_1_1"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c4">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E3:29_1_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <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/><a NAME="E4:29_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:29_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A17</font></span>: 
<font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><a NAME="E6:29_1_1"/>
 <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</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="#E1:29_1_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="pre_poly.html#T51" title="PRE_POLY:th.51">PRE_POLY:51</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E7:29_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:29_1_1"/>
 <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="pre_poly.html#T52" title="PRE_POLY:th.52">PRE_POLY:52</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="29_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_2"/>
<font color="Maroon" title="c1">n</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">x</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">n9</font> = <font color="Maroon" title="c1">n</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">x9</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c13">n9</font>,<font color="Maroon" title="c2">L</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">b1</font> = <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c16">b2</font> = <font color="Maroon" title="c4">b2</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c13">n9</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">sb2</font> =  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>), <font color="Maroon" title="c18">sb1</font> =  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font></span>)</span>) as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c13">n9</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">u</font> = <font color="Maroon" title="c11">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">n9</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c20">yb2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E8:29_1_2"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">yb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">sb2</font>
 <span class="kw">and </span><br/><a NAME="E9:29_1_2"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
 <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c16">b2</font>,<font color="Maroon" title="c14">x9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c20">yb2</font>
 <span class="kw">and </span><br/><a NAME="E10:29_1_2"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">yb2</font> holds <br/><font color="Maroon" title="c20">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">sb2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c17">sb2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#D2" target="_self" title="POLYNOM2:def.2">Def2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">yb1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E12:29_1_2"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">yb1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">sb1</font>
 <span class="kw">and </span><br/><a NAME="E13:29_1_2"/><span class="lab"><font color="Green" title="E43">A22</font></span>: 
 <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c14">x9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c21">yb1</font>
 <span class="kw">and </span><br/><a NAME="E14:29_1_2"/><span class="lab"><font color="Green" title="E44">A23</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">yb1</font> holds <br/><font color="Maroon" title="c21">yb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#D2" target="_self" title="POLYNOM2:def.2">Def2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">ytmp</font> =  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>);<br/><span class="lab"><font color="Green" title="E45">A24</font></span>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c15">b1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#T13" target="_self" title="POLYNOM2:th.13">Th15</a>, <a class="ref" href="pre_poly.html#T11" title="PRE_POLY:th.11">PRE_POLY:11</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c16">b2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="card_2.html#T41" title="CARD_2:th.41">CARD_2:41</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">sb2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#T13" target="_self" title="POLYNOM2:th.13">Th15</a>, <a class="ref" href="pre_poly.html#T11" title="PRE_POLY:th.11">PRE_POLY:11</a></span>
;<br/><a NAME="E16:29_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A25</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">yb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_1_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E12:29_1_2"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></span>;<br/><a NAME="E17:29_1_2"/><span class="lab"><font color="Green" title="E47">A26</font></span>: 
<font color="Maroon" title="c17">sb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E13:29"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E20:29"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="polynom2.html#T11" target="_self" title="POLYNOM2:th.11">Th13</a>, <a class="ref" href="polynom2.html#T13" target="_self" title="POLYNOM2:th.13">Th15</a></span>;<br/><a NAME="E18:29_1_2"/><span class="lab"><font color="Green" title="E48">A27</font></span>: 
 for <font color="Olive" title="b1">i</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">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">yb2</font> holds <br/><font color="Maroon" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">i</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="c23">i</font> &amp; <font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">yb2</font> implies <font color="Maroon" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29_1_2_2"/><span class="lab"><font color="Green" title="E49">A28</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font>
 <span class="kw">and </span><br/><a NAME="E2:29_1_2_2"/><span class="lab"><font color="Green" title="E50">A29</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">yb2</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="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span><br/>

<a NAME="E3:29_1_2_2"/><span class="lab"><font color="Green" title="E51">A30</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">yb2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E2:29_1_2_2"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:29_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A31</font></span>: 
<font color="Maroon" title="c20">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p3">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:29_1_2"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="txt" href="#E10:29_1_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E2:29_1_2_2"><span class="lab"><font color="Green" title="E50">A29</font></span></a></span>;<br/>
<a NAME="E5:29_1_2_2"/><span class="lab"><font color="Green" title="E53">A32</font></span>: 
<font color="Maroon" title="c23">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="c20">yb2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_2_2"><span class="lab"><font color="Green" title="E51">A30</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="E6:29_1_2_2"/>
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_1_2"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E2:29_1_2_2"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E7:29_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A33</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:29_1_2_2"/><span class="lab"><font color="Green" title="E55">A34</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E9:29_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A35</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 is    <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_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="E1:29_1_2_2_1_1"/>
( <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or <font color="Maroon" title="c23">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:29_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E57">A36</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span><br/><div class="add"><a NAME="E2:29_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E58">A37</font></span>: 
<font color="Maroon" title="c17">sb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c19">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:29_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A38</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">sb2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A39</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">sb2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">sb2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E3:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E59">A38</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
;<br/><a NAME="E5:29_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E61">A40</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c16">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T37" title="PRE_POLY:th.37">PRE_POLY:37</a></span>;<br/><a NAME="E6:29_1_2_2_1_1_1"/>
<font color="Maroon" title="c17">sb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c16">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E2:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:29_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A41</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c17">sb2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E5:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E61">A40</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c17">sb2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c17">sb2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c16">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E7:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E62">A41</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
;<br/><a NAME="E9:29_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A42</font></span>: 
<font color="Maroon" title="c20">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">u</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_1_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E2:29_1_2_2"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="txt" href="#E3:29_1_2_2"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="txt" href="#E1:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="txt" href="#E4:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E60">A39</font></span></a></span>;<br/><a NAME="E10:29_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E64">A43</font></span>: 
<font color="Maroon" title="c12">k9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">yb1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:29"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E8:29_1_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E12:29_1_2"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E2:29_1_2_2"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="txt" href="#E1:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1_2_2"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c20">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E20:29"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E1:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="txt" href="#E10:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E64">A43</font></span></a>, <a class="txt" href="#E9:29_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E63">A42</font></span></a>, <a class="ref" href="finseq_5.html#T73" title="FINSEQ_5:th.73">FINSEQ_5:73</a></span>
<br/>.= 
<font color="Maroon" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_2_2"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 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="29_1_2_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:29_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E57">A44</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">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" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span><br/><div class="add"><a NAME="E2:29_1_2_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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="c18">sb1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></span>;<br/><a NAME="E3:29_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A45</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_2_1_1_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="E1:29_1_2_2_1_1_2_1_1"/>
( <font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> or <font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_2_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:29_1_2_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E59">A46</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">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" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span><br/><div class="add"><a NAME="E2:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A47</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:29"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A48</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:29_1_2_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A49</font></span>: 
<font color="Maroon" title="c21">yb1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">k9</font></span>)</span> is   <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="E5:29_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">k9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">yb1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E20:29"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E12:29_1_2"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="txt" href="#E4:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E7:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A50</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/><a NAME="E8:29_1_2_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E64">A51</font></span>: 
<font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">k9</font></span>)</span> is   <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="E9:29_1_2_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E65">A52</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">sb1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E10:29_1_2_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E66">A53</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">yb2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E8:29_1_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E1:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A54</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">yb1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_1_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E12:29_1_2"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E12:29_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">sb1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_1_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E10:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A53</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">sb1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E14:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A55</font></span>: 
<font color="Maroon" title="c23">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="c18">sb1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E15:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A56</font></span>: 
<font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">sb1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E16:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A57</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E14:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E9:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><div><span class="lab"><font color="Green" title="E71">A58</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2_1_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_1_2_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</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/><a NAME="E2:29_1_2_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E3:29_1_2_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">sb1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:29_1_2_2_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E21:29"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</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="E18:29_1_2_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E72">A59</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E19:29_1_2_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E73">A60</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E20:29_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A56</font></span></a>, <a class="txt" href="#E9:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a></span>;<br/><a NAME="E21:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E22:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A61</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><a NAME="E23:29_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E2:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E24:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E8:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E25:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A62</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/><a NAME="E26:29_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <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="#E16:29"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E1:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E27:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A63</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E3:29_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E28:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A64</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E29:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A65</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E27:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E19:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A66</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A65</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><a NAME="E31:29_1_2_2_1_1_2_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E32:29_1_2_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A67</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E28:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="txt" href="#E19:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c16">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c16">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c16">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></span>
<br/>.= 
<font color="Maroon" title="c15">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E17:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c15">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A68</font></span>: <font color="Maroon" title="c20">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> = 
<font color="Maroon" title="c21">yb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_1_2"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E3:29_1_2_2"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="txt" href="#E4:29_1_2_2"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="txt" href="#E11:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="txt" href="#E30:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A66</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></span>
;<br/><span class="kw">thus </span><font color="Maroon" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> = 
<font color="Maroon" title="c20">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_2_2"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1_2_2"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E34:29_1_2_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 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="29_1_2_2_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:29_1_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E59">A69</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">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" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c24">i1</font> = <font color="Maroon" title="c23">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_1_2_2"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><a NAME="E3:29_1_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E60">A70</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 ;<br/><a NAME="E4:29_1_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E61">A71</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">sb1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E5:29_1_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E62">A72</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">sb2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_1_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E2:29_1_2_2"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E6:29_1_2_2_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:29"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E1:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A69</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A73</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_2_2"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="txt" href="#E3:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E60">A70</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">sb1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E5:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A72</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A74</font></span>: 
<font color="Maroon" title="c24">i1</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="c18">sb1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E10:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A75</font></span>: 
<font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">sb1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E11:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A76</font></span>: 
<font color="Maroon" title="c24">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E9:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="txt" href="#E4:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A71</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><a NAME="E12:29_1_2_2_1_1_2_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E13:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A77</font></span>: 
<font color="Maroon" title="c24">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="txt" href="#E10:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A75</font></span></a>, <a class="txt" href="#E4:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A71</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><div><span class="lab"><font color="Green" title="E68">A78</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_1_2_2_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</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/><a NAME="E2:29_1_2_2_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E3:29_1_2_2_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">sb1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:29_1_2_2_1_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E21:29"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</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="E15:29_1_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E69">A79</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E16:29_1_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A80</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E17:29_1_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E71">A81</font></span>: 
<font color="Maroon" title="c24">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 ;<br/><a NAME="E18:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A82</font></span>: 
<font color="Maroon" title="c12">k9</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="c24">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A69</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E19:29_1_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E73">A83</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_1_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E1:29_1_2_2"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E2:29_1_2_2"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="txt" href="#E3:29_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E20:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A84</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E21:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A85</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E19:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A83</font></span></a>, <a class="txt" href="#E16:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A80</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A86</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A85</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A83</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E5:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A72</font></span></a>, <a class="txt" href="#E17:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A81</font></span></a>, <a class="txt" href="#E18:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A82</font></span></a>, <a class="ref" href="finseq_5.html#T74" title="FINSEQ_5:th.74">FINSEQ_5:74</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A76</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A76</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><a NAME="E23:29_1_2_2_1_1_2_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E24:29_1_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A87</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A83</font></span></a>, <a class="txt" href="#E20:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A84</font></span></a>, <a class="txt" href="#E16:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A80</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c16">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A87</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c16">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c18">sb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c19">u</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A83</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c16">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E5:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A72</font></span></a>, <a class="txt" href="#E17:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A81</font></span></a>, <a class="txt" href="#E18:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A82</font></span></a>, <a class="ref" href="finseq_5.html#T74" title="FINSEQ_5:th.74">FINSEQ_5:74</a></span>
<br/>.= 
<font color="Maroon" title="c15">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E14:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A78</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c15">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">sb1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A74</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">i1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A77</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <font color="Maroon" title="c18">sb1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A77</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A88</font></span>: <font color="Maroon" title="c20">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> = 
<font color="Maroon" title="c21">yb1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29_1_2"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="txt" href="#E14:29_1_2"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E4:29_1_2_2"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="txt" href="#E7:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E63">A73</font></span></a>, <a class="txt" href="#E5:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A72</font></span></a>, <a class="txt" href="#E22:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A86</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29_1_2"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="txt" href="#E15:29_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E5:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A72</font></span></a>, <a class="txt" href="#E15:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A79</font></span></a>, <a class="txt" href="#E18:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A82</font></span></a>, <a class="ref" href="finseq_5.html#T74" title="FINSEQ_5:th.74">FINSEQ_5:74</a></span>
;<br/><span class="kw">thus </span><font color="Maroon" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> = 
<font color="Maroon" title="c20">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_2_2"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1_2_2"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E26:29_1_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A88</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:29_1_2_2_1_1_2"/>
<font color="Maroon" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</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/></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="E11:29_1_2_2"/>
<font color="Maroon" title="c20">yb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</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/>


</div><span class="kw">end;</span></div> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k9</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k9</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T5" title="GROUP_4:th.5">GROUP_4:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T5" title="GROUP_4:th.5">GROUP_4:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k9</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">yb1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c12">k9</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T5" title="GROUP_4:th.5">GROUP_4:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c21">yb1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T8" title="RFINSEQ:th.8">RFINSEQ:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c14">x9</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:29_1_2"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="ref" href="group_4.html#T9" title="GROUP_4:th.9">GROUP_4:9</a></span>
;<br/><a NAME="E20:29_1_2"/><span class="kw">then </span>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c21">yb1</font>,<font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c14">x9</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D4" title="FINSEQ_5:def.4">FINSEQ_5:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E21:29_1_2"/>
 <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c8">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_2"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E16:29_1_2"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E18:29_1_2"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</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>
