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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#T8" onmouseover="rs('rfunct_1/T8')" onmouseout="rh()">RFUNCT_1:8</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#T8" onmouseover="rs('rfunct_1/T8')" onmouseout="rh()">RFUNCT_1:8</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> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</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="rvsum_1.html#T22" onmouseover="rs('rvsum_1/T22')" onmouseout="rh()">RVSUM_1:22</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#T16" onmouseover="rs('rvsum_1/T16')" onmouseout="rh()">RVSUM_1:16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:4"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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