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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Olive" title="b1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">p</font> = <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>;<br/>
<font color="Maroon" title="c4">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> = 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T58" target="_self" title="EUCLID_8:th.58">Th58</a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:77"/>
<font color="Maroon" title="c4">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
 ; <a class="txt" 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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