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<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="vectmetr.html#V4" target="_self">strict</a> <a href="vectmetr.html#NM2">RealLinearMetrSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_1_2"/><i><font color="Green">E11</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:28_1_2"/><i><font color="Green">E12</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R1">=</a>  <a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:28_1_2"/><i><font color="Green">E13</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:28_1_2"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> holds the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K7">+</a> <font color="Olive">b<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:28_1_2"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:28_1_2"/><i><font color="Green">E16</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E7:28_1_2"/><i><font color="Green">E17</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a>  <a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E8:28_1_2"/><i><font color="Green">E18</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E9:28_1_2"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> holds the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K7">+</a> <font color="Olive">b<sub>2</sub></font>
 <b>and </b><br/><a NAME="E10:28_1_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="vectmetr.html#E1:28_1_2"><i><font color="Green">E11</font></i></a></i>;<br/><b>thus </b>the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_2"><i><font color="Green">E14</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E9:28_1_2"><i><font color="Green">E19</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E12:28_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:28_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="vectmetr.html#E6:28_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="vectmetr.html#E1:28_1_2"><i><font color="Green">E11</font></i></a></i>;<br/><b>thus </b>the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E5:28_1_2"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E10:28_1_2"><i><font color="Green">E20</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:28_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:28_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="vectmetr.html#E2:28_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="vectmetr.html#E3:28_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="vectmetr.html#E6:28_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="vectmetr.html#E7:28_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="vectmetr.html#E8:28_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="vectmetr.html#E12:28_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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