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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> implies 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">X</font><br/> for <font color="Olive" title="b2">S</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b2">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> holds <br/>for <font color="Olive" title="b3">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b3">H</font> &amp; ( for <font color="Olive" title="b4">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">S</font> holds <br/><font color="Olive" title="b3">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b4">y</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  ) holds <br/> <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Olive" title="b2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Olive" title="b3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )</span><br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E13">A1</font></i>: 
( the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">S</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b2">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> holds <br/>for <font color="Olive" title="b3">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b3">H</font> &amp; ( for <font color="Olive" title="b4">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">S</font> holds <br/><font color="Olive" title="b3">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b4">y</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  ) holds <br/> <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Olive" title="b2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Olive" title="b3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span><br/>

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> holds <br/>for <font color="Olive" title="b2">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">H</font> &amp; ( for <font color="Olive" title="b3">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  ) holds <br/> <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Olive" title="b1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Olive" title="b2">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span><br/><b>let </b><font color="Maroon" title="c3">S</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c3">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies for <font color="Olive" title="b1">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">H</font> &amp; ( 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="c1">X</font>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> holds <br/><font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  ) holds <br/> <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Olive" title="b1">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )</span><br/>



<b>assume </b><a NAME="E2:24"/><i><font color="Green" title="E14">A2</font></i>: 
not <font color="Maroon" title="c3">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">H</font> &amp; ( 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="c1">X</font>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> holds <br/><font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  ) holds <br/> <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Olive" title="b1">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span><br/>

<b>let </b><font color="Maroon" title="c4">H</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">H</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> holds <br/><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  ) implies  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c4">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )</span><br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">H</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> holds <br/><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c4">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><a NAME="E1:24_1"/><i><font color="Green" title="E16">A4</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:24_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E1:24_1_1"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c2">x</font>;<br/>
<a NAME="E3:24_1_1"/>
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c6">y1</font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">a</font> = the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y2</font> = <font color="Maroon" title="c5">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E1:24_1_1"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<a NAME="E6:24_1_1"/>
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">X</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">y2</font>
 
;<br/>
<b>hence </b><a NAME="E7:24_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E2:24_1"/>
 ex <font color="Olive" title="b1">F0</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> st <br/>for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E1:24_1"><i><font color="Green" title="E16">A4</font></i></a>);<br/></i><b>then consider </b><font color="Maroon" title="c5">F0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:24_1"/><i><font color="Green" title="E17">A6</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c5">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><a NAME="E5:24_1"/><i><font color="Green" title="E18">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c5">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">y</font> )</span><br/><b>assume </b><a NAME="E1:24_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">y</font></span><br/><b>thus </b><font color="Maroon" title="c5">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> = 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E4:24_1"><i><font color="Green" title="E17">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c6">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:24_1"/><i><font color="Green" title="E19">A8</font></i>: 
( <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">F</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> holds <br/><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">y</font> ) )
 <b>by </b><i><a class="txt" href="#E5:24_1"><i><font color="Green" title="E18">A7</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c7">z</font> =  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font>;<br/><a NAME="E9:24_1"/>
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c4">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> 
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E2:24"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E3:24"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E8:24_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="bhsp_5.html#T11" target="_self" title="BHSP_5:th.11">Th11</a></i>;<br/><a NAME="E10:24_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E2:24"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E3:24"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E8:24_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="bhsp_5.html#T12" target="_self" title="BHSP_5:th.12">Th12</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p4">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">F</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T18" title="BHSP_1:th.18">BHSP_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c4">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E2:24"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E3:24"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E8:24_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="bhsp_5.html#T12" target="_self" title="BHSP_5:th.12">Th12</a></i>
;<br/><b>hence </b><a NAME="E12:24_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c4">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E5:24"/><b>then </b><i><font color="Green" title="E16">A9</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c4">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:24"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E7:24"/><b>then </b>
 <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c4">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="#E5:24"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:24"/>
 <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c4">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <b>by </b><i><a class="ref" href="bhsp_1.html#D4" title="BHSP_1:def.4">BHSP_1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
