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<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>4</sub></font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E5:13"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E6:13"/><i><font color="Green">E13</font></i>: 
 <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4">halfline</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E2:13"><i><font color="Green">E9</font></i></a>, <a class="txt" href="brouwer.html#E3:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="brouwer.html#E4:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="brouwer.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E7:13"/><b>then </b>
 <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K4">halfline</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:13"/><i><font color="Green">E14</font></i>: 
 <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E10:13"/><i><font color="Green">E15</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="topreal9.html#T26">TOPREAL9:26</a></i>;<br/>
<i><font color="Green">E16</font></i>:  <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E9:13"><i><font color="Green">E14</font></i></a>, <a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="jordan2c.html#T9">JORDAN2C:9</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>
;<br/>
<a NAME="E12:13"/><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="brouwer.html#E1:13"><i><font color="Green">E8</font></i></a>, <a class="txt" href="brouwer.html#E4:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="euclid.html#T19">EUCLID:19</a></i>;<br/>
<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><a NAME="E2:13_2"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/><b>hence </b><a NAME="E3:13_2"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E12:13"><i><font color="Green">E17</font></i></a>, <a class="ref" href="euclid.html#D5">EUCLID:def 5</a>, <a class="ref" href="square_1.html#T82">SQUARE_1:82</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green">E19</font></i>:  <a href="quin_1.html#K1">delta</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quin_1.html#D1">QUIN_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
4 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E15:13"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E16:13"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E2:13"><i><font color="Green">E9</font></i></a>, <a class="ref" href="topreal9.html#T8">TOPREAL9:8</a></i>;<br/>
<a NAME="E17:13"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D5">EUCLID:def 5</a></i>;<br/>
<a NAME="E18:13"/><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="euclid.html#T12">EUCLID:12</a></i>;<br/>
<a NAME="E19:13"/>
<font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E1:13"><i><font color="Green">E8</font></i></a>, <a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/>
<a NAME="E20:13"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#D6">TOPRNS_1:def 6</a></i>;<br/>
<a NAME="E21:13"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D5">EUCLID:def 5</a></i>;<br/>
<a NAME="E22:13"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/>
<a NAME="E23:13"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E21:13"><i><font color="Green">E24</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E24:13"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/>
<a NAME="E25:13"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E17:13"><i><font color="Green">E21</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E26:13"/>
<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E1:13"><i><font color="Green">E8</font></i></a>, <a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/>
<a NAME="E27:13"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#D6">TOPRNS_1:def 6</a></i>;<br/>
<a NAME="E28:13"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E16:13"><i><font color="Green">E20</font></i></a>, <a class="txt" href="brouwer.html#E17:13"><i><font color="Green">E21</font></i></a>, <a class="txt" href="brouwer.html#E18:13"><i><font color="Green">E22</font></i></a>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E29:13"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="brouwer.html#E17:13"><i><font color="Green">E21</font></i></a>, <a class="txt" href="brouwer.html#E25:13"><i><font color="Green">E26</font></i></a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E30:13"/><i><font color="Green">E30</font></i>: 
 <a href="quin_1.html#K1">delta</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="quin_1.html#D1">QUIN_1:def 1</a></i>;<br/>
<a NAME="E31:13"/>
4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/>
<a NAME="E32:13"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="brouwer.html#E29:13"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/>
<a NAME="E33:13"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="brouwer.html#E32:13"><i><font color="Green">E31</font></i></a>, <a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/>
<a NAME="E34:13"/>
<font color="Maroon">c<sub>18</sub></font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E33</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="toprns_1.html#T8">TOPRNS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T161">COMPLEX1:161</a></i>
;<br/>
<a NAME="E36:13"/>
 <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E6:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>3</sub></font> = 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="txt" href="brouwer.html#E9:13"><i><font color="Green">E14</font></i></a>, <a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T51">EUCLID:51</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="topreal9.html#T1">TOPREAL9:1</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T48">EUCLID:48</a></i>
;<br/>
<b>then </b><i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T67">EUCLID_2:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T41">EUCLID_2:41</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E23:13"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E25:13"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E20:13"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E35:13"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K18">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E27:13"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="brouwer.html#E38:13"><i><font color="Green">E34</font></i></a></i>
;<br/>
<a NAME="E40:13"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E41:13"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="polyeq_1.html#K3">Polynom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>,<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="polyeq_1.html#D2">POLYEQ_1:def 2</a></i>;<br/>
<a NAME="E42:13"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green">E18</font></i></a>, <a class="txt" href="brouwer.html#E29:13"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/>
<a NAME="E43:13"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="real_1.html#K1">-</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E44:13"/>
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E43:13"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>then </b><i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 4</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E14:13"><i><font color="Green">E19</font></i></a>, <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#T85">SQUARE_1:85</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E5:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
;<br/>
<a NAME="E46:13"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green">E18</font></i></a>, <a class="txt" href="brouwer.html#E30:13"><i><font color="Green">E30</font></i></a>, <a class="txt" href="brouwer.html#E41:13"><i><font color="Green">E35</font></i></a>, <a class="txt" href="brouwer.html#E33:13"><i><font color="Green">E32</font></i></a>, <a class="ref" href="polyeq_1.html#T5">POLYEQ_1:5</a></i>;<br/>
<a NAME="E47:13"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <a href="polyeq_1.html#K3">Polynom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="polyeq_1.html#K5">Quard</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>thus </b> <a href="polyeq_1.html#K3">Polynom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>,<font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyeq_1.html#D2">POLYEQ_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green">E18</font></i></a>, <a class="txt" href="brouwer.html#E30:13"><i><font color="Green">E30</font></i></a>, <a class="txt" href="brouwer.html#E33:13"><i><font color="Green">E32</font></i></a>, <a class="ref" href="quin_1.html#T16">QUIN_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="polyeq_1.html#K5">Quard</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="polyeq_1.html#D3">POLYEQ_1:def 3</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E48:13"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="polyeq_1.html#T11">POLYEQ_1:11</a></i>;<br/>
<a NAME="E49:13"/><i><font color="Green">E40</font></i>: 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_10"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>19</sub></font> =  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:13_10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:13_10"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E49:13"><i><font color="Green">E40</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E3:13_10"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E4:13_10"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:13_10"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E30:13"><i><font color="Green">E30</font></i></a>, <a class="txt" href="brouwer.html#E33:13"><i><font color="Green">E32</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><a NAME="E6:13_10"/><b>then </b>
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E7:13_10"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E4:13_10"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_11"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_11"/>
( <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="brouwer.html#E28:13"><i><font color="Green">E28</font></i></a>, <a class="txt" href="brouwer.html#E17:13"><i><font color="Green">E21</font></i></a>, <a class="txt" href="brouwer.html#E25:13"><i><font color="Green">E26</font></i></a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_11_1"><b>suppose </b></a><a NAME="E1:13_11_1"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:13_11_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="brouwer.html#E13:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="real_2.html#T128">REAL_2:128</a></i>;<br/><b>hence </b><a NAME="E3:13_11_1"/>
 <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E50:13"><i><font color="Green">E41</font></i></a>, <a class="txt" href="brouwer.html#E46:13"><i><font color="Green">E38</font></i></a>, <a class="txt" href="brouwer.html#E45:13"><i><font color="Green">E37</font></i></a>, <a class="txt" href="brouwer.html#E10:13"><i><font color="Green">E15</font></i></a>, <a class="txt" href="brouwer.html#E9:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="brouwer.html#E48:13"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T135">XREAL_1:135</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_11_2"><b>suppose </b></a><a NAME="E1:13_11_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_11_2"/>
 <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E46:13"><i><font color="Green">E38</font></i></a>, <a class="txt" href="brouwer.html#E45:13"><i><font color="Green">E37</font></i></a>, <a class="txt" href="brouwer.html#E9:13"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_11_3"><b>suppose </b></a><a NAME="E1:13_11_3"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_11_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_11_3_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_11_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_11_3_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then </b> <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E11:13"><i><font color="Green">E16</font></i></a>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/><b>hence </b><a NAME="E3:13_11_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E2:13"><i><font color="Green">E9</font></i></a>, <a class="txt" href="brouwer.html#E3:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="brouwer.html#E4:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="brouwer.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_11_3_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_11_3_1_2"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> or 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_11_3_1_2_1"><b>suppose </b></a><a NAME="E1:13_11_3_1_2_1"/><i><font color="Green">E45</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span>
 ;<br/><div class="add"><a NAME="E2:13_11_3_1_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><a NAME="E3:13_11_3_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E1:13_11_3_1_2_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="brouwer.html#E1:13_11_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="brouwer.html#E30:13"><i><font color="Green">E30</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><b>hence </b><a NAME="E4:13_11_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E1:13_11_3_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="brouwer.html#E2:13_11_3_1_2_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="brouwer.html#E10:13"><i><font color="Green">E15</font></i></a>, <a class="txt" href="brouwer.html#E49:13"><i><font color="Green">E40</font></i></a>, <a class="ref" href="real_2.html#T128">REAL_2:128</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_11_3_1_2_2"><b>suppose </b></a><a NAME="E1:13_11_3_1_2_2"/><i><font color="Green">E45</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E1:13_11_3_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="brouwer.html#E1:13_11_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="brouwer.html#E30:13"><i><font color="Green">E30</font></i></a>, <a class="ref" href="square_1.html#T90">SQUARE_1:90</a></i>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E3:13_11_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E1:13_11_3_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="brouwer.html#E2:13_11_3_1"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:13_11_3"/>
 <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E46:13"><i><font color="Green">E38</font></i></a>, <a class="txt" href="brouwer.html#E45:13"><i><font color="Green">E37</font></i></a>, <a class="txt" href="brouwer.html#E9:13"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
