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<span class="kw">let </span><font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span> iff  ex <font color="Olive" title="b2">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b4">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b4">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) ) &amp; (  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b5">x3</font>, <font color="Olive" title="b6">y3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b5">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b2">y1</font>,<font color="Olive" title="b4">y2</font>,<font color="Olive" title="b6">y3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b6">y3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font><br/>  for <font color="Olive" title="b4">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds   <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b4">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b4">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:33"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span> iff  ex <font color="Olive" title="b2">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b4">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b4">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b5">x3</font>, <font color="Olive" title="b6">y3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b5">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b2">y1</font>,<font color="Olive" title="b4">y2</font>,<font color="Olive" title="b6">y3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b6">y3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font><br/>  for <font color="Olive" title="b4">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds   <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b4">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b4">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) ) ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span> iff  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> implies <font color="Maroon" title="c4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:33_1_1"/>
<font color="Maroon" title="c4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x1x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>, <font color="Maroon" title="c6">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E3:33_1_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><font color="Maroon" title="c5">x1x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T18" onmouseover="rs('prvect_3/T18')" onmouseout="rh()">PRVECT_3:18</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c8">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E5:33_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c5">x1x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T18" onmouseover="rs('prvect_3/T18')" onmouseout="rh()">PRVECT_3:18</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c7">x1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b2">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c10">x2</font> = <font color="Maroon" title="c8">x2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c11">x3</font> = <font color="Maroon" title="c6">x3</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">thus </span><a NAME="E6:33_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E5:33_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:33_1"/>
(  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> implies <font color="Maroon" title="c4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font><br/>  for <font color="Olive" title="b4">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds   <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b4">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b4">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) ) ) )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">y1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font>, <font color="Olive" title="b4">y3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b4">y3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">y3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">y2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x3</font>, <font color="Olive" title="b2">y3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c7">y2</font>,<font color="Olive" title="b2">y3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">x3</font>, <font color="Maroon" title="c9">y3</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c8">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">y3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><a NAME="E1:33_2"/>
<span class="p1"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x2</font></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c7">y2</font></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y2</font></span>)</span></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T18" onmouseover="rs('prvect_3/T18')" onmouseout="rh()">PRVECT_3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E2:33_2"/>
<span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c8">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">y3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T18" onmouseover="rs('prvect_3/T18')" onmouseout="rh()">PRVECT_3:18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:33"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_4.html#K6" title="PRVECT_4:func.6">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K6" title="PRVECT_4:func.6">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font><br/>  for <font color="Olive" title="b4">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds   <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b4">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b4">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:33"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font><br/>  for <font color="Olive" title="b4">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds   <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> ) &amp; (  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b4">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b4">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) ) ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b2">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font><br/>  for <font color="Olive" title="b3">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">x3</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/>







<span class="kw">thus </span><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> = 
<span class="p1"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x3</font></span>)</span></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T18" onmouseover="rs('prvect_3/T18')" onmouseout="rh()">PRVECT_3:18</a></span>
<br/>.= 
<span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T18" onmouseover="rs('prvect_3/T18')" onmouseout="rh()">PRVECT_3:18</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b4">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b4">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b2">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds   <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds   <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">x3</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><br/><span class="kw">thus </span> <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" onmouseover="rs('rlvect_1/T16')" onmouseout="rh()">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" onmouseover="rs('rlvect_1/T16')" onmouseout="rh()">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" onmouseover="rs('rlvect_1/T16')" onmouseout="rh()">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">x3</font></span>)</span></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" onmouseover="rs('rlvect_1/T16')" onmouseout="rh()">RLVECT_1:16</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c4">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b2">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b3">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x2</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b3">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">y1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b2">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x3</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b2">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">z1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">z1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">z1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c7">v10</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<span class="kw"> such that </span><br/><a NAME="E7:33"/><span class="lab"><font color="Green" title="E8">A11</font></span>: 
( <font color="Maroon" title="c7">v10</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p3"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">z1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="prvect_3.html#K8" title="PRVECT_3:func.8">prod_NORM</a> (<span class="p2"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<font color="Maroon" title="c3">G</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span>,<font color="Maroon" title="c6">z1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c7">v10</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D6" onmouseover="rs('prvect_3/D6')" onmouseout="rh()">PRVECT_3:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">v20</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<span class="kw"> such that </span><br/><a NAME="E9:33"/><span class="lab"><font color="Green" title="E9">A12</font></span>: 
( <font color="Maroon" title="c8">v20</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="prvect_3.html#K8" title="PRVECT_3:func.8">prod_NORM</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">v20</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D6" onmouseover="rs('prvect_3/D6')" onmouseout="rh()">PRVECT_3:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">v1</font> = <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">z1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T104" onmouseover="rs('finseq_2/T104')" onmouseout="rh()">FINSEQ_2:104</a></span>;<br/>
<a NAME="E11:33"/><span class="lab"><font color="Green" title="E10">A14</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c8">v20</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T86" onmouseover="rs('rvsum_1/T86')" onmouseout="rh()">RVSUM_1:86</a></span>;<br/>
<span class="lab"><font color="Green" title="E11">A15</font></span>: <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c8">v20</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E10">A14</font></span></a>, <a class="ref" href="square_1.html#D2" onmouseover="rs('square_1/D2')" onmouseout="rh()">SQUARE_1:def 2</a>, <a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E9">A12</font></span></a></span>
<br/>.= 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E9">A12</font></span></a>, <a class="ref" href="topreal6.html#T11" onmouseover="rs('topreal6/T11')" onmouseout="rh()">TOPREAL6:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T77" onmouseover="rs('rvsum_1/T77')" onmouseout="rh()">RVSUM_1:77</a></span>
;<br/>
<span class="lab"><font color="Green" title="E12">A16</font></span>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c7">v10</font></span>)</span> = 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p4"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">z1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T11" onmouseover="rs('topreal6/T11')" onmouseout="rh()">TOPREAL6:11</a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E8">A11</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">z1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="ref" href="rvsum_1.html#T77" onmouseover="rs('rvsum_1/T77')" onmouseout="rh()">RVSUM_1:77</a></span>
;<br/>
<a NAME="E14:33"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c7">v10</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">v1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="ref" href="borsuk_7.html#T17" onmouseover="rs('borsuk_7/T17')" onmouseout="rh()">BORSUK_7:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:33"/>
( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">z1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3 st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">y1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>,<span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">z1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E8">A11</font></span></a>, <a class="txt" href="#E13:33"><span class="lab"><font color="Green" title="E12">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
