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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="polynom1.html#V2" target="_self">finite-Support</a> );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E2:187_1_1"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="polynom1.html#V2" target="_self">finite-Support</a> <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E4:187_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K22" target="_self">+</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>3</sub></font> );<br/>
<div><i><font color="Green">E104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:187_1_1_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="polynom1.html#V2" target="_self">finite-Support</a> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:187_1_1_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="polynom1.html#V2" target="_self">finite-Support</a> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="polynom1.html#E3:187_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="polynom1.html#E5:187_1_1_1"><i><font color="Green">E106</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K22" target="_self">+</a> <font color="Maroon">c<sub>13</sub></font>;<br/><a NAME="E7:187_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K22" target="_self">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K22" target="_self">+</a> <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font>;<br/><b>thus </b><a NAME="E9:187_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>16</sub></font>]
 <b>by </b><i><a class="txt" href="polynom1.html#E3:187_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="polynom1.html#E5:187_1_1_1"><i><font color="Green">E106</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E8:187_1_1"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="polynom1.html#E6:187_1_1"><i><font color="Green">E104</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>3</sub></font> );<br/>
<div><i><font color="Green">E106</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:187_1_1_2"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="polynom1.html#V2" target="_self">finite-Support</a> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:187_1_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="polynom1.html#V2" target="_self">finite-Support</a> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="polynom1.html#E3:187_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="polynom1.html#E5:187_1_1_2"><i><font color="Green">E108</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">c<sub>14</sub></font>;<br/><a NAME="E7:187_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font>;<br/><b>thus </b><a NAME="E9:187_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font>]
 <b>by </b><i><a class="txt" href="polynom1.html#E3:187_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="polynom1.html#E5:187_1_1_2"><i><font color="Green">E108</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E11:187_1_1"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="polynom1.html#E9:187_1_1"><i><font color="Green">E106</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_3.html#K5">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM6">Function</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:187_1_1_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom1.html#E1:187_1_1_3"><i><font color="Green">E108</font></i></a></i>;<br/><a NAME="E3:187_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E4:187_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E1:187_1_1_3"><i><font color="Green">E108</font></i></a>, <a class="ref" href="polynom1.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:187_1_1"/><b>then </b>
 <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E18:187_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> is <a href="polynom1.html#V2" target="_self">finite-Support</a>
 
<b>by </b><i><a class="ref" href="polynom1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E19:187_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM6">Function</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:187_1_1_4_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font> ;<br/><b>assume </b><a NAME="E3:187_1_1_4_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then </b><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E5:187_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E1:187_1_1_4_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="polynom1.html#D9" target="_self">Def9</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:187_1_1_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:187_1_1_4"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E4:187_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E5:187_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E6:187_1_1_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="polynom1.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E26:187_1_1"/><b>then </b>
 <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E27:187_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> is <a href="polynom1.html#V2" target="_self">finite-Support</a>
 
<b>by </b><i><a class="ref" href="polynom1.html#D10" target="_self">Def10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> =  <a href="vectsp_1.html#G3">doubleLoopStr</a>(# <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>12</sub></font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V3">strict</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>thus </b><a NAME="E30:187_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:187_1_1_5_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:187_1_1_5_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> is <a href="polynom1.html#V2" target="_self">finite-Support</a> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="polynom1.html#E1:187_1_1_5_1"><i><font color="Green">E108</font></i></a></i>;<br/><b>thus </b><a NAME="E5:187_1_1_5_1"/>
<font color="Maroon">c<sub>19</sub></font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E4:187_1_1_5_1"><i><font color="Green">E109</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:187_1_1_5"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:187_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E4:187_1_1_5"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:187_1_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="polynom1.html#E2:187_1_1_5"><i><font color="Green">E108</font></i></a></i>;<br/>


</div><b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>18</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:187_1_1_6"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:187_1_1_6"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> )
 <b>and </b><br/><a NAME="E4:187_1_1_6"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K22" target="_self">+</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E8:187_1_1"><i><font color="Green">E105</font></i></a></i>;<br/><b>thus </b><a NAME="E5:187_1_1_6"/>
<font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="polynom1.html#K22" target="_self">+</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:187_1_1_6"><i><font color="Green">E108</font></i></a>, <a class="txt" href="polynom1.html#E3:187_1_1_6"><i><font color="Green">E109</font></i></a>, <a class="txt" href="polynom1.html#E4:187_1_1_6"><i><font color="Green">E110</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>18</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:187_1_1_7"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:187_1_1_7"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> )
 <b>and </b><br/><a NAME="E4:187_1_1_7"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E11:187_1_1"><i><font color="Green">E107</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>19</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>20</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>

<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="polynom1.html#E1:187_1_1_7"><i><font color="Green">E108</font></i></a>, <a class="txt" href="polynom1.html#E3:187_1_1_7"><i><font color="Green">E109</font></i></a>, <a class="txt" href="polynom1.html#E4:187_1_1_7"><i><font color="Green">E110</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>thus </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_3.html#K5">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
 <a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>

;<br/>

<b>thus </b> <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_3.html#K5">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
 <a href="polynom1.html#K27" target="_self">1_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>

;<br/>


</div>
