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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="gtarski1.html#V9" title="GTARSKI1:attr.9">satisfying_Tarski-model</a>   <a href="gtarski1.html#L1" title="GTARSKI1:struct.1">TarskiGeometryStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">A9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b3">x</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> holds <br/> (  <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">A9</font> iff (  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">A9</font> &amp; <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">A9</font> <a href="gtarski4.html#R7" title="GTARSKI4:pred.7">Is</a> <font color="Olive" title="b3">x</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">A9</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> holds <br/> (  <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">A9</font> iff (  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> <a href="gtarski4.html#R7" title="GTARSKI4:pred.7">Is</a> <font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A9</font> iff (  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> <a href="gtarski4.html#R7" title="GTARSKI4:pred.7">Is</a> <font color="Maroon" title="c4">x</font> ) )</span><br/>







<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> <a href="gtarski4.html#R7" title="GTARSKI4:pred.7">Is</a> <font color="Maroon" title="c4">x</font> implies  <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A9</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> <a href="gtarski4.html#R7" title="GTARSKI4:pred.7">Is</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">hence </span><a NAME="E2:35_1"/>
 <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> <a href="gtarski4.html#R7" title="GTARSKI4:pred.7">Is</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">hence </span><a NAME="E3:35_1"/>
<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> <a href="gtarski4.html#R7" title="GTARSKI4:pred.7">Is</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="gtarski4.html#T25" target="_self" onmouseover="rs('gtarski4/T25')" onmouseout="rh()">Satz8p14p1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:35"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
(  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> <a href="gtarski4.html#R7" title="GTARSKI4:pred.7">Is</a> <font color="Maroon" title="c4">x</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A9</font></span><br/>

<a NAME="E3:35"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st  <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">A9</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:35"/>
 <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="gtarski3.html#T88" onmouseover="rs('gtarski3/T88')" onmouseout="rh()">GTARSKI3:88</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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