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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font>, <font color="Olive" title="b3">R</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ec_pf_1.html#K3" title="EC_PF_1:func.3">ProjCo</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span>  st <font color="Olive" title="b1">P</font> <a href="ec_pf_1.html#R1" title="EC_PF_1:pred.1">_EQ_</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Olive" title="b2">Q</font> <a href="ec_pf_1.html#R1" title="EC_PF_1:pred.1">_EQ_</a> <font color="Olive" title="b3">R</font> holds <br/><font color="Olive" title="b1">P</font> <a href="ec_pf_1.html#R1" title="EC_PF_1:pred.1">_EQ_</a> <font color="Olive" title="b3">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font>, <font color="Maroon" title="c4">R</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ec_pf_1.html#K3" title="EC_PF_1:func.3">ProjCo</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> <a href="ec_pf_1.html#R1" title="EC_PF_1:pred.1">_EQ_</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c3">Q</font> <a href="ec_pf_1.html#R1" title="EC_PF_1:pred.1">_EQ_</a> <font color="Maroon" title="c4">R</font> implies <font color="Maroon" title="c2">P</font> <a href="ec_pf_1.html#R1" title="EC_PF_1:pred.1">_EQ_</a> <font color="Maroon" title="c4">R</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E52">AS</font></span>: 
( <font color="Maroon" title="c2">P</font> <a href="ec_pf_1.html#R1" title="EC_PF_1:pred.1">_EQ_</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c3">Q</font> <a href="ec_pf_1.html#R1" title="EC_PF_1:pred.1">_EQ_</a> <font color="Maroon" title="c4">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">P</font> <a href="ec_pf_1.html#R1" title="EC_PF_1:pred.1">_EQ_</a> <font color="Maroon" title="c4">R</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:70"/><span class="lab"><font color="Green" title="E53">P1</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Maroon" title="c2">P</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> &amp; <font color="Maroon" title="c2">P</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span> &amp; <font color="Maroon" title="c2">P</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#D10" target="_self" title="EC_PF_1:def.10">DefEQV</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:70"/><span class="lab"><font color="Green" title="E54">Q1</font></span>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Maroon" title="c3">Q</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> &amp; <font color="Maroon" title="c3">Q</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span> &amp; <font color="Maroon" title="c3">Q</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#D10" target="_self" title="EC_PF_1:def.10">DefEQV</a>, <a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E52">AS</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">b</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ec_pf_1.html#D10" target="_self" title="EC_PF_1:def.10">EC_PF_1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Maroon" title="c2">P</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> &amp; <font color="Maroon" title="c2">P</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span> &amp; <font color="Maroon" title="c2">P</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E6:70"/>
( <font color="Maroon" title="c7">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Maroon" title="c2">P</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> &amp; <font color="Maroon" title="c2">P</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span> &amp; <font color="Maroon" title="c2">P</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E53">P1</font></span></a>, <a class="txt" href="#E5:70"><span class="lab"><font color="Green" title="E54">Q1</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a>, <a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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