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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="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">p</font>, <font color="Olive" title="b3">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><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="Olive" title="b1">L</font><br/> for <font color="Olive" title="b5">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st not <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Olive" title="b2">p</font> &amp;  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Olive" title="b2">p</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="b5">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b6">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b6">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b</font> holds <br/><font color="Olive" title="b3">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b6">b'</font> ) holds <br/> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Olive" title="b3">q</font>,<font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b5">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b4">x</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  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="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">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="b4">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st not <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Olive" title="b1">p</font> &amp;  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Olive" title="b1">p</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">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b5">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b5">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b</font> holds <br/><font color="Olive" title="b2">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b5">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b5">b'</font> ) holds <br/> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b3">x</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font> &amp;  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</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="b2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b3">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b3">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b3">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b3">b'</font> ) holds <br/> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">x</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font> &amp;  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</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">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b2">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b2">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b2">b'</font> ) holds <br/> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b1">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font> &amp;  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</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="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b1">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b1">b'</font> ) implies  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span> )</span><br/>









<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E35">A1</font></i>: 
( not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font> &amp;  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</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="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b1">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b1">b'</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c7">sq</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">sp</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span>;<br/>
<a NAME="E2:47"/><i><font color="Green" title="E36">A2</font></i>: 
(  <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font> )
 
<b>by </b><i><a class="ref" href="polynom2.html#T20" target="_self" title="POLYNOM2:th.20">Th20</a></i>;<br/>
<a NAME="E3:47"/>
<font color="Maroon" title="c6">b</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="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:47"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:47"/><b>then </b>
<font color="Maroon" title="c6">b</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:47"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:47"/><i><font color="Green" title="E37">A3</font></i>: 
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E8:47"/><i><font color="Green" title="E38">A4</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="#E7:47"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E9:47"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:47"/><b>then </b><i><font color="Green" title="E39">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E7:47"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:47"/><b>then </b>
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="c9">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">k'</font> = <font color="Maroon" title="c9">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c6">b</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E14:47"/><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c10">k'</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="c9">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="E15:47"/><b>then </b><i><font color="Green" title="E41">A7</font></i>: 
 <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c11">b</font>
 
<b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E2:47"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E7:47"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E8:47"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="polynom2.html#T13" target="_self" title="POLYNOM2:th.13">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">yq</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><b> such that </b><br/><a NAME="E17:47"/><i><font color="Green" title="E42">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">yq</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> &amp;  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c12">yq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="c12">yq</font> holds <br/><font color="Maroon" title="c12">yq</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom2.html#D4" target="_self" title="POLYNOM2:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">yp</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><b> such that </b><br/><a NAME="E19:47"/><i><font color="Green" title="E43">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">yp</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> &amp;  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c13">yp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="c13">yp</font> holds <br/><font color="Maroon" title="c13">yp</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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom2.html#D4" target="_self" title="POLYNOM2:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c11">b</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>set </b><font color="Maroon" title="c15">ytmp</font> =  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>;<br/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">yp</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k'</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&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="c13">yp</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">k'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">yp</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k'</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">yp</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">k'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">yp</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">yp</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">k'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">yp</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">yp</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">k'</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">yp</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k'</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="c13">yp</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">k'</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c13">yp</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="rlvect_1.html#T61" title="RLVECT_1:th.61">RLVECT_1:61</a></i>
;<br/>
<a NAME="E22:47"/><b>then </b><i><font color="Green" title="E44">A10</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D4" title="FINSEQ_5:def.4">FINSEQ_5:def 4</a></i>;<br/>
<i><font color="Green" title="E45">A11</font></i>: <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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="polynom2.html#T20" target="_self" title="POLYNOM2:th.20">Th20</a>, <a class="ref" href="triang_1.html#T9" title="TRIANG_1:th.9">TRIANG_1:9</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom2.html#T20" target="_self" title="POLYNOM2:th.20">Th20</a>, <a class="ref" href="triang_1.html#T9" title="TRIANG_1:th.9">TRIANG_1:9</a></i>
;<br/>
<a NAME="E24:47"/><b>then </b><i><font color="Green" title="E46">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">yq</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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>;<br/>
<a NAME="E25:47"/>
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="c12">yq</font> holds <br/><font color="Maroon" title="c12">yq</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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></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="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">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="c12">yq</font> implies <font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> )</span><br/>

<b>assume </b><a NAME="E1:47_3"/><i><font color="Green" title="E47">A13</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">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="c12">yq</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span><br/>

<a NAME="E2:47_3"/><b>then </b>
( <font color="Maroon" title="c16">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="c12">yq</font></span>)</span> &amp; <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E24:47"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:47_3"/><b>then </b><i><font color="Green" title="E48">A14</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">yq</font> &amp; <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_3_1"/>
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> or <font color="Maroon" title="c16">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_1_1"><b>suppose </b></a><a NAME="E1:47_3_1_1"/><i><font color="Green" title="E49">A15</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span><br/><div class="add"><a NAME="E2:47_3_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:47_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="#E7:47"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E4:47_3_1_1"/><b>then </b><i><font color="Green" title="E50">A16</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:47"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green" title="E51">A17</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font>
<b>by </b><i><a class="txt" href="#E7:47"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E4:47_3_1_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><i><font color="Green" title="E52">A18</font></i>: <font color="Maroon" title="c12">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:47"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="#E1:47_3_1_1"><i><font color="Green" title="E49">A15</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:47"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E5:47_3_1_1"><i><font color="Green" title="E51">A17</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><a NAME="E7:47_3_1_1"/><i><font color="Green" title="E53">A19</font></i>: 
<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> <font color="Maroon" title="c13">yp</font>
 <b>by </b><i><a class="txt" href="#E10:47"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="#E14:47"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E23:47"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E3:47_3"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E14:47"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="#E1:47_3_1_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="#E7:47_3_1_1"><i><font color="Green" title="E53">A19</font></i></a>, <a class="ref" href="finseq_5.html#T76" title="FINSEQ_5:th.76">FINSEQ_5:76</a></i>
<br/>.= 
<font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E3:47_3"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="#E6:47_3_1_1"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_1_2"><b>suppose </b></a><a NAME="E1:47_3_1_2"/><i><font color="Green" title="E49">A20</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span><br/><div class="add"><a NAME="E2:47_3_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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>;<br/><a NAME="E3:47_3_1_2"/><b>then </b><i><font color="Green" title="E50">A21</font></i>: 
 <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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_3_1_2_1_1"/>
( <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> or <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font> )
 </span><b>by </b><i><a class="txt" href="#E1:47_3_1_2"><i><font color="Green" title="E49">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:47_3_1_2_1_1_1"/><i><font color="Green" title="E51">A22</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span><br/><div class="add"><a NAME="E2:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E52">A23</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k'</font>
 <b>by </b><i><a class="txt" href="#E14:47"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E53">A24</font></i>: 
<font color="Maroon" title="c16">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="c10">k'</font>
 <b>by </b><i><a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:47_3_1_2_1_1_1"/><i><font color="Green" title="E54">A25</font></i>: 
<font color="Maroon" title="c9">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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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
 <b>by </b><i><a class="txt" href="#E10:47"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="#E23:47"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E55">A26</font></i>: 
<font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:47_3_1_2_1_1_1"><i><font color="Green" title="E52">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:47_3_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <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="c10">k'</font></span>)</span> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></i>;<br/><a NAME="E7:47_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <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="c10">k'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:47_3_1_2_1_1_1"><i><font color="Green" title="E53">A24</font></i></a>, <a class="txt" href="#E4:47_3_1_2_1_1_1"><i><font color="Green" title="E54">A25</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><a NAME="E8:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E56">A27</font></i>: 
<font color="Maroon" title="c16">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"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k'</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E9:47_3_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E10:47_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E5:47_3_1_2_1_1_1"><i><font color="Green" title="E55">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E57">A28</font></i>: 
<font color="Maroon" title="c16">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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="#E3:47_3_1_2"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E58">A29</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:47_3_1_2_1_1_1"/><i><font color="Green" title="E59">A30</font></i>: 
 <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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E14:47_3_1_2_1_1_1"/><i><font color="Green" title="E60">A31</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> is   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E15:47_3_1_2_1_1_1"/>
<font color="Maroon" title="c16">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="#E5:47_3_1_2_1_1_1"><i><font color="Green" title="E55">A26</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E61">A32</font></i>: 
<font color="Maroon" title="c16">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E17:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E62">A33</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E18:47_3_1_2_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E19:47_3_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="#E17:47_3_1_2_1_1_1"><i><font color="Green" title="E62">A33</font></i></a></i>;<br/><a NAME="E20:47_3_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E63">A34</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:47_3_1_2_1_1_1"><i><font color="Green" title="E61">A32</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><div><i><font color="Green" title="E64">A35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_3_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:47_3_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="#E16:47_3_1_2_1_1_1"><i><font color="Green" title="E61">A32</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E3:47_3_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">b</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:47_3_1_2_1_1_1"><i><font color="Green" title="E61">A32</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:47_3_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E2:47"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E23:47_3_1_2_1_1_1"/>
<font color="Maroon" title="c10">k'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="#E14:47"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></i>;<br/><a NAME="E24:47_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">k'</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="c12">yq</font>
 <b>by </b><i><a class="txt" href="#E10:47"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E25:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E65">A36</font></i>: 
<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> <font color="Maroon" title="c13">yp</font>
 <b>by </b><i><a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E23:47"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E26:47_3_1_2_1_1_1"/>
<font color="Maroon" title="c16">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="c12">yq</font>
 <b>by </b><i><a class="txt" href="#E10:47"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E1:47_3_1_2_1_1_1"><i><font color="Green" title="E51">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E27:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E66">A37</font></i>: 
<font color="Maroon" title="c16">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="c13">yp</font>
 <b>by </b><i><a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E23:47"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E28:47_3_1_2_1_1_1"/><i><font color="Green" title="E67">A38</font></i>: 
<font color="Maroon" title="c16">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="c10">k'</font>
 <b>by </b><i><a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="#E2:47_3_1_2_1_1_1"><i><font color="Green" title="E52">A23</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E29:47_3_1_2_1_1_1"/>
<font color="Maroon" title="c13">yp</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="c10">k'</font></span>)</span> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></i>;<br/><a NAME="E30:47_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">yp</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="c10">k'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E25:47_3_1_2_1_1_1"><i><font color="Green" title="E65">A36</font></i></a>, <a class="txt" href="#E28:47_3_1_2_1_1_1"><i><font color="Green" title="E67">A38</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><a NAME="E31:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E68">A39</font></i>: 
<font color="Maroon" title="c16">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="c13">yp</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k'</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E32:47_3_1_2_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E33:47_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E69">A40</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:47_3_1_2_1_1_1"><i><font color="Green" title="E57">A28</font></i></a>, <a class="txt" href="#E12:47_3_1_2_1_1_1"><i><font color="Green" title="E58">A29</font></i></a>, <a class="txt" href="#E13:47_3_1_2_1_1_1"><i><font color="Green" title="E59">A30</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green" title="E70">A41</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E33:47_3_1_2_1_1_1"><i><font color="Green" title="E69">A40</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:47_3_1_2_1_1_1"><i><font color="Green" title="E57">A28</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E8:47_3_1_2_1_1_1"><i><font color="Green" title="E56">A27</font></i></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></i>
<br/>.= 
<font color="Maroon" title="c3">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E14:47_3_1_2_1_1_1"><i><font color="Green" title="E60">A31</font></i></a>, <a class="txt" href="#E22:47_3_1_2_1_1_1"><i><font color="Green" title="E64">A35</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">p</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E16:47_3_1_2_1_1_1"><i><font color="Green" title="E61">A32</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E21:47_3_1_2_1_1_1"><i><font color="Green" title="E63">A34</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E21:47_3_1_2_1_1_1"><i><font color="Green" title="E63">A34</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><i><font color="Green" title="E71">A42</font></i>: <font color="Maroon" title="c12">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="#E11:47_3_1_2_1_1_1"><i><font color="Green" title="E57">A28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E15:47"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="#E8:47_3_1_2_1_1_1"><i><font color="Green" title="E56">A27</font></i></a>, <a class="txt" href="#E34:47_3_1_2_1_1_1"><i><font color="Green" title="E70">A41</font></i></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></i>
<br/>.= 
<font color="Maroon" title="c13">yp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="#E11:47_3_1_2_1_1_1"><i><font color="Green" title="E57">A28</font></i></a>, <a class="txt" href="#E27:47_3_1_2_1_1_1"><i><font color="Green" title="E66">A37</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E31:47_3_1_2_1_1_1"><i><font color="Green" title="E68">A39</font></i></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></i>
;<br/><b>thus </b><font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<font color="Maroon" title="c12">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E3:47_3"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E3:47_3"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="#E35:47_3_1_2_1_1_1"><i><font color="Green" title="E71">A42</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:47_3_1_2_1_1_2"/><i><font color="Green" title="E51">A43</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span><br/><div class="add"><a NAME="E2:47_3_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">k'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">i1</font> = <font color="Maroon" title="c16">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E4:47_3_1_2_1_1_2"/><i><font color="Green" title="E52">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c16">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="E5:47_3_1_2_1_1_2"/><i><font color="Green" title="E53">A45</font></i>: 
<font color="Maroon" title="c16">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">yp</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
 <b>by </b><i><a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E23:47"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E6:47_3_1_2_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c10">k'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:47_3_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="txt" href="#E1:47_3_1_2_1_1_2"><i><font color="Green" title="E51">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:47_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E54">A46</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i1</font>
 <b>by </b><i><a class="txt" href="#E4:47_3_1_2_1_1_2"><i><font color="Green" title="E52">A44</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:47_3_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E5:47_3_1_2_1_1_2"><i><font color="Green" title="E53">A45</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:47_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E55">A47</font></i>: 
<font color="Maroon" title="c17">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> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E11:47_3_1_2_1_1_2"/><i><font color="Green" title="E56">A48</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i1</font> is   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><div><i><font color="Green" title="E57">A49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_3_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:47_3_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="#E10:47_3_1_2_1_1_2"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E3:47_3_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">b</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:47_3_1_2_1_1_2"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:47_3_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E2:47"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E13:47_3_1_2_1_1_2"/><i><font color="Green" title="E58">A50</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:47_3_1_2_1_1_2"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:47_3_1_2_1_1_2"/><i><font color="Green" title="E59">A51</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E15:47_3_1_2_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E16:47_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E60">A52</font></i>: 
<font color="Maroon" title="c17">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:47_3_1_2_1_1_2"><i><font color="Green" title="E55">A47</font></i></a>, <a class="txt" href="#E13:47_3_1_2_1_1_2"><i><font color="Green" title="E58">A50</font></i></a>, <a class="txt" href="#E14:47_3_1_2_1_1_2"><i><font color="Green" title="E59">A51</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E17:47_3_1_2_1_1_2"/><i><font color="Green" title="E61">A53</font></i>: 
<font color="Maroon" title="c16">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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E23:47"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="#E3:47_3_1_2"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E18:47_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E62">A54</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E19:47_3_1_2_1_1_2"/><i><font color="Green" title="E63">A55</font></i>: 
 <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> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E20:47_3_1_2_1_1_2"/><i><font color="Green" title="E64">A56</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E21:47_3_1_2_1_1_2"/><i><font color="Green" title="E65">A57</font></i>: 
<font color="Maroon" title="c10">k'</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="c17">i1</font>
 <b>by </b><i><a class="txt" href="#E1:47_3_1_2_1_1_2"><i><font color="Green" title="E51">A43</font></i></a>, <a class="txt" href="#E4:47_3_1_2_1_1_2"><i><font color="Green" title="E52">A44</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E22:47_3_1_2_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E23:47_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E66">A58</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E17:47_3_1_2_1_1_2"><i><font color="Green" title="E61">A53</font></i></a>, <a class="txt" href="#E18:47_3_1_2_1_1_2"><i><font color="Green" title="E62">A54</font></i></a>, <a class="txt" href="#E19:47_3_1_2_1_1_2"><i><font color="Green" title="E63">A55</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green" title="E67">A59</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E23:47_3_1_2_1_1_2"><i><font color="Green" title="E66">A58</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E17:47_3_1_2_1_1_2"><i><font color="Green" title="E61">A53</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E5:47_3_1_2_1_1_2"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="#E20:47_3_1_2_1_1_2"><i><font color="Green" title="E64">A56</font></i></a>, <a class="txt" href="#E21:47_3_1_2_1_1_2"><i><font color="Green" title="E65">A57</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
<br/>.= 
<font color="Maroon" title="c3">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E11:47_3_1_2_1_1_2"><i><font color="Green" title="E56">A48</font></i></a>, <a class="txt" href="#E12:47_3_1_2_1_1_2"><i><font color="Green" title="E57">A49</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">p</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E10:47_3_1_2_1_1_2"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i1</font>
<b>by </b><i><a class="txt" href="#E16:47_3_1_2_1_1_2"><i><font color="Green" title="E60">A52</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i1</font>
<b>by </b><i><a class="txt" href="#E16:47_3_1_2_1_1_2"><i><font color="Green" title="E60">A52</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><i><font color="Green" title="E68">A60</font></i>: <font color="Maroon" title="c12">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E1:47_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="#E17:47_3_1_2_1_1_2"><i><font color="Green" title="E61">A53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i1</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E15:47"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E5:47_3_1_2_1_1_2"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="#E20:47_3_1_2_1_1_2"><i><font color="Green" title="E64">A56</font></i></a>, <a class="txt" href="#E21:47_3_1_2_1_1_2"><i><font color="Green" title="E65">A57</font></i></a>, <a class="txt" href="#E24:47_3_1_2_1_1_2"><i><font color="Green" title="E67">A59</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
<br/>.= 
<font color="Maroon" title="c13">yp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i1</font>
<b>by </b><i><a class="txt" href="#E19:47"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="#E5:47_3_1_2_1_1_2"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="#E8:47_3_1_2_1_1_2"><i><font color="Green" title="E54">A46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E5:47_3_1_2_1_1_2"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="#E20:47_3_1_2_1_1_2"><i><font color="Green" title="E64">A56</font></i></a>, <a class="txt" href="#E21:47_3_1_2_1_1_2"><i><font color="Green" title="E65">A57</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>
;<br/><b>thus </b><font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<font color="Maroon" title="c12">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E3:47_3"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E3:47_3"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="#E25:47_3_1_2_1_1_2"><i><font color="Green" title="E68">A60</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:47_3_1_2"/>
<font color="Maroon" title="c12">yq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">yp</font>,<font color="Maroon" title="c10">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E26:47"/>
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E17:47"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E22:47"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="#E24:47"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
