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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c1">f</font> holds <br/> <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K29" title="SEQ_4:func.29">Incr</a> <font color="Maroon" title="c1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c1">f</font> implies  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K29" title="SEQ_4:func.29">Incr</a> <font color="Maroon" title="c1">f</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K29" title="SEQ_4:func.29">Incr</a> <font color="Maroon" title="c1">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">F</font> =  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c2">A</font> as   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T12" target="_self" title="JORDAN1H:th.12">Th12</a></span>;<br/>
<a NAME="E3:21"/>
 <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>  <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T11" target="_self" title="JORDAN1H:th.11">Th11</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></span>;<br/>
<a NAME="E4:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</a></span>;<br/>
<a NAME="E5:21"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="jordan1h.html#T11" target="_self" title="JORDAN1H:th.11">Th11</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a>, <a class="ref" href="pre_poly.html#T11" title="PRE_POLY:th.11">PRE_POLY:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21"/>
 <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K29" title="SEQ_4:func.29">Incr</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="seq_4.html#D25" title="SEQ_4:def.25">SEQ_4:def 25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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