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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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> 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  <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b2">f</font> <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="c2">y</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b2">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">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> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">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  <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b1">f</font> <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="c2">y</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">f</font> <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="c4">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="c2">y</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>







<a NAME="E1:8"/>
 <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1" title="EUCLID:def.1">EUCLID:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">f</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<a NAME="E3:8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T54" title="EUCLID:th.54">EUCLID:54</a></i>;<br/>
<b>hence </b><a NAME="E4:8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">f</font> <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="c4">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="c2">y</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <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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