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Step * 1 1 of Lemma mk-inner-product-space_wf


1. ss : self:"Point":Type
        "#":{s:self."Point" ⟶ self."Point" ⟶ ℙ| ∀x:self."Point". (¬(s x x))}  ⋂ x:Atom ⟶ if x =a "#or"
                                                                                       then ∀x,y,z:self."Point".
                                                                                              ((self."#" x y)
                                                                                              ⇒ ((self."#" x z)
                                                                                                 ∨ (self."#" y z)))
                                                                                       else Top
                                                                                       fi 
2. ss ∈ record(x.Top)
3. xx : Top
4. xxx : Top
5. xxxx : Top
6. ss."Point" ∈ Type
7. ss."#" ∈ {s:ss."Point" ⟶ ss."Point" ⟶ ℙ| ∀x:ss."Point". (¬(s x x))} 
8. ss."#or" ∈ ∀x,y,z:ss."Point".  ((ss."#" x y) ⇒ ((ss."#" x z) ∨ (ss."#" y z)))
⊢ ss["ip" := xx]["positive" := xxx]["perp" := xxxx] ∈ record(x.Top)
BY
{ (RepUR ``record record-update`` 0 THEN Auto) }

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Latex:

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Latex:

1.  ss  :  self:"Point":Type
                "\#":\{s:self."Point"  {}\mrightarrow{}  self."Point"  {}\mrightarrow{}  \mBbbP{}|  \mforall{}x:self."Point".  (\mneg{}(s  x  x))\}    \mcap{}  x:Atom
                {}\mrightarrow{}  if  x  =a  "\#or"
                      then  \mforall{}x,y,z:self."Point".    ((self."\#"  x  y)  {}\mRightarrow{}  ((self."\#"  x  z)  \mvee{}  (self."\#"  y  z)))
                      else  Top
                      fi 
2.  ss  \mmember{}  record(x.Top)
3.  xx  :  Top
4.  xxx  :  Top
5.  xxxx  :  Top
6.  ss."Point"  \mmember{}  Type
7.  ss."\#"  \mmember{}  \{s:ss."Point"  {}\mrightarrow{}  ss."Point"  {}\mrightarrow{}  \mBbbP{}|  \mforall{}x:ss."Point".  (\mneg{}(s  x  x))\} 
8.  ss."\#or"  \mmember{}  \mforall{}x,y,z:ss."Point".    ((ss."\#"  x  y)  {}\mRightarrow{}  ((ss."\#"  x  z)  \mvee{}  (ss."\#"  y  z)))
\mvdash{}  ss["ip"  :=  xx]["positive"  :=  xxx]["perp"  :=  xxxx]  \mmember{}  record(x.Top)


By

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Latex:
(RepUR  ``record  record-update``  0  THEN  Auto)

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