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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">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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">consider </span><font color="Maroon" title="c4">I</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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 class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
( <font color="Maroon" title="c4">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b1">x</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">y</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">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <font color="Maroon" title="c4">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> ) &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <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> holds  <font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">v</font> being   <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><br/>  for <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><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>)</span> &amp; (  for <font color="Olive" title="b1">v</font> being   <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> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</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="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_4.html#T10" target="_self" onmouseover="rs('prvect_4/T10')" onmouseout="rh()">Th15</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:36"/>
 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">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</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="c5">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> implies <font color="Maroon" title="c5">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:36_1_1"/>
<font color="Maroon" title="c5">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><a NAME="E2:36_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E3:36_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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 class="kw"> such that </span><br/><a NAME="E5:36_1_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" onmouseover="rs('funct_2/T113')" onmouseout="rh()">FUNCT_2:113</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>, <font color="Maroon" title="c9">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="E7:36_1_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c6">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="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">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="prvect_4.html#T9" target="_self" onmouseover="rs('prvect_4/T9')" onmouseout="rh()">Th14</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c10">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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c11">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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c12">x3</font> = <font color="Maroon" title="c9">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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">x2</font>,<font color="Maroon" title="c12">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><a NAME="E8:36_1_1"/>
<font color="Maroon" title="c4">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">x2</font>,<font color="Maroon" title="c12">x3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">x2</font>,<font color="Maroon" title="c12">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:36_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">x2</font>,<font color="Maroon" title="c12">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E7:36_1_1"><span class="lab"><font color="Green" title="E10">A4</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="E2:36_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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> implies <font color="Maroon" title="c5">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</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="E4:36"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
 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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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="36_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b4">y3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">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#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">x2</font>, <font color="Maroon" title="c8">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">y2</font>,<font color="Olive" title="b2">y3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">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#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">x3</font>, <font color="Maroon" title="c10">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">y2</font>,<font color="Maroon" title="c10">y3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">y3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/>





<a NAME="E1:36_2"/><span class="lab"><font color="Green" title="E10">A9</font></span>: 
<span class="p1"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">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="c6">y1</font>,<font color="Maroon" title="c8">y2</font>,<font color="Maroon" title="c10">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="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">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_4.html#T9" target="_self" onmouseover="rs('prvect_4/T9')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E2:36_2"/><span class="lab"><font color="Green" title="E11">A10</font></span>: 
<font color="Maroon" title="c4">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">y3</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">y3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/>
<a NAME="E3:36_2"/>
( <font color="Maroon" title="c4">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">x3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> &amp; <font color="Maroon" title="c4">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">y2</font>,<font color="Maroon" title="c10">y3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">y2</font>,<font color="Maroon" title="c10">y3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:36_2"/>
<span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">y2</font>,<font color="Maroon" title="c10">y3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">y3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E1:36_2"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E2:36_2"><span class="lab"><font color="Green" title="E11">A10</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="E5:36"/><span class="lab"><font color="Green" title="E10">A11</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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="36_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:36_3"/>
<font color="Maroon" title="c4">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<span class="p1">(<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="p1">(<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="p1">(<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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:36_3"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</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>

<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><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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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="c5">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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="c5">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#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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="c5">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="c6">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#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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="c5">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="c6">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="c7">x3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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="c5">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="c6">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="c7">x3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> = 
<span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">x1</font></span>)</span></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> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">x3</font></span>)</span></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E9">A7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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"><font color="Maroon" title="c6">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">x3</font></span>)</span></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" onmouseover="rs('rlvect_1/D10')" onmouseout="rh()">RLVECT_1:def 10</a></span>
<br/>.= 
<span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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"><font color="Maroon" title="c7">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">x3</font></span>)</span></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" onmouseover="rs('rlvect_1/D10')" onmouseout="rh()">RLVECT_1:def 10</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</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="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="ref" href="rlvect_1.html#D10" onmouseover="rs('rlvect_1/D10')" onmouseout="rh()">RLVECT_1:def 10</a></span>
;<br/><span class="kw">hence </span><a NAME="E2:36_4"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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="c5">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="c6">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="c7">x3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" onmouseover="rs('rlvect_1/D10')" onmouseout="rh()">RLVECT_1:def 10</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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#K8" title="PRVECT_4:func.8">&lt;*</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#K8" title="PRVECT_4:func.8">*&gt;</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="c5">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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="c5">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#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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="c5">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="c6">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#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</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="c5">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="c6">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="c7">x3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">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="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span></span><br/><a NAME="E1:36_5"/><span class="lab"><font color="Green" title="E11">A14</font></span>: 
<span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/><font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="prvect_4.html#K7" title="PRVECT_4:func.7">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span></span>)</span> = 
<font color="Maroon" title="c4">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x3</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_4.html#T9" target="_self" onmouseover="rs('prvect_4/T9')" onmouseout="rh()">Th14</a></span>
<br/>.= 
<span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:36_5"/>
<font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x3</font></span>)</span></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_5"><span class="lab"><font color="Green" title="E11">A14</font></span></a>, <a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</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">let </span><font color="Maroon" title="c5">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="c5">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="c5">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="c6">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 <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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="c5">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="c6">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="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="c5">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="c6">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="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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="c7">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"> ( <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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="c5">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="c6">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="Maroon" title="c7">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="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="c5">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="c6">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="Maroon" title="c7">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="lab"><font color="Green" title="E11">A16</font></span>: <font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2: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">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K7" title="PRVECT_4:func.7">]</a></span> = 
<font color="Maroon" title="c4">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>)

<br/>.= 
<span class="p1"><a href="prvect_4.html#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>
;<br/>
<a NAME="E9:36"/>
<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="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">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="c5">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="c6">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="Maroon" title="c7">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_4.html#T9" target="_self" onmouseover="rs('prvect_4/T9')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:36"/>
<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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="c5">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="c6">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="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E11">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">  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="c5">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="c6">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="Maroon" title="c7">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="c8">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<span class="kw"> such that </span><br/><a NAME="E12:36"/><span class="lab"><font color="Green" title="E12">A17</font></span>: 
( <font color="Maroon" title="c8">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="c5">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="c6">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="Maroon" title="c7">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="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">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="Maroon" title="c8">w</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_4.html#T9" target="_self" onmouseover="rs('prvect_4/T9')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">w</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="c8">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="c5">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="c6">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="Maroon" title="c7">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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="Maroon" title="c8">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E13:36"/>
<font color="Maroon" title="c8">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="c5">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="c6">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="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E12">A17</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"> <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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="Maroon" title="c8">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/>

<span class="kw">thus </span><a NAME="E14:36"/>
<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#K8" title="PRVECT_4:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="prvect_4.html#K8" title="PRVECT_4:func.8">*&gt;</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="Maroon" title="c8">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="#E2:36"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E12">A17</font></span></a>, <a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E11">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>
