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<b>let </b><font color="Maroon" title="c1">a1</font>, <font color="Maroon" title="c2">a2</font>, <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">x1</font>, <font color="Olive" title="b3">x2</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="Olive" title="b1">n</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b2">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b3">x2</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b2">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b3">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b2">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b3">x2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">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">x1</font>, <font color="Olive" title="b2">x2</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="c5">n</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b2">x2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</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="c5">n</font>; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><br/>













<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T23" target="_self" title="EUCLIDLP:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</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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