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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b3">c</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">c</font></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">a2</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">b2</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c7">c2</font> = <font color="Maroon" title="c4">c</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">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>  ;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a2</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">b2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">c2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T19" title="RFUNCT_1:th.19">RFUNCT_1:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">c2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T19" title="RFUNCT_1:th.19">RFUNCT_1:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">c2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T40" title="RVSUM_1:th.40">RVSUM_1:40</a></span>
<br/>.= 
<font color="Maroon" title="c5">a2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">c2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T33" title="RVSUM_1:th.33">RVSUM_1:33</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">c</font>
 ; <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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