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<b>let </b><font color="Maroon" title="c1">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b3">g</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b2">g</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">g</font></span>)</span></span><br/>





<b>thus </b><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c4">g</font></span>)</span> = 
<font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K7" title="RVSUM_1:func.7">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52" title="RVSUM_1:th.52">RVSUM_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K7" title="RVSUM_1:func.7">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T73" title="RVSUM_1:th.73">RVSUM_1:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">g</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_1.html#T6" target="_self" title="TOPALG_1:th.6">Th6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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