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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E43">XYZbS3</font></span>: <a NAME="T39"><span class="comment"><font color="firebrick">:: REALALG3:39</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">F</font> being   <a href="realalg1.html#V14" title="REALALG1:attr.14">ordered</a>  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b3">P</font> being   <a href="realalg1.html#NM2" title="REALALG1:NM.2">Ordering</a> of <font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">E</font>  st <font color="Olive" title="b4">a</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">F</font> holds <br/> for <font color="Olive" title="b5">f</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <font color="Olive">b<sub>3</sub></font> <a href="realalg3.html#V2" title="REALALG3:attr.2">-quadratic</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>  ex <font color="Olive" title="b6">g1</font>, <font color="Olive" title="b7">g2</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <br/>(  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b6">g1</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"><a href="realalg3.html#K3" title="REALALG3:func.3">@</a> (<span class="p3">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<span class="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Olive" title="b4">a</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b7">g2</font></span>)</span></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b8">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b8">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b6">g1</font> holds <br/> ex <font color="Olive" title="b9">b</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> ex <font color="Olive" title="b10">c1</font>, <font color="Olive" title="b11">c2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <br/>( <font color="Olive" title="b9">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b10">c1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b11">c2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b6">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b10">c1</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b11">c2</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="realalg3.html#K3" title="REALALG3:func.3">@</a> (<span class="p5">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<span class="p0"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Olive" title="b4">a</font>)</span>)</span> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> ) ) &amp; (  for <font color="Olive" title="b8">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b8">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b7">g2</font> holds <br/> ex <font color="Olive" title="b9">b</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> ex <font color="Olive" title="b10">c1</font>, <font color="Olive" title="b11">c2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Olive" title="b1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> st <br/>( <font color="Olive" title="b9">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b10">c1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b11">c2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b7">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b9">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b10">c1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b11">c2</font> ) ) )</div></div>
