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Step * 1 1 of Lemma funinv-funinv


1. n : ℕ
2. f : {f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} 
3. inv(f) ∈ {f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} 
4. ∀a1,a2:ℕn.  (((inv(f) a1) = (inv(f) a2) ∈ ℕn) ⇒ (a1 = a2 ∈ ℕn))
5. Inj(ℕn;ℕn;f)
6. inv(inv(f)) ∈ {f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} 
7. Inj(ℕn;ℕn;inv(inv(f)))
8. x : ℕn
⊢ (inv(f) (inv(inv(f)) x)) = (inv(f) (f x)) ∈ ℕn
BY
{ ((InstLemma `funinv-property` [⌜n⌝;⌜inv(f)⌝;⌜x⌝]⋅ THENA Auto)
   THEN InstLemma `funinv-property` [⌜n⌝;⌜f⌝;⌜x⌝]⋅
   THEN Auto) }

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Latex:

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Latex:

1.  n  :  \mBbbN{}
2.  f  :  \{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\} 
3.  inv(f)  \mmember{}  \{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\} 
4.  \mforall{}a1,a2:\mBbbN{}n.    (((inv(f)  a1)  =  (inv(f)  a2))  {}\mRightarrow{}  (a1  =  a2))
5.  Inj(\mBbbN{}n;\mBbbN{}n;f)
6.  inv(inv(f))  \mmember{}  \{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\} 
7.  Inj(\mBbbN{}n;\mBbbN{}n;inv(inv(f)))
8.  x  :  \mBbbN{}n
\mvdash{}  (inv(f)  (inv(inv(f))  x))  =  (inv(f)  (f  x))


By

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Latex:
((InstLemma  `funinv-property`  [\mkleeneopen{}n\mkleeneclose{};\mkleeneopen{}inv(f)\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  InstLemma  `funinv-property`  [\mkleeneopen{}n\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{}]\mcdot{}
  THEN  Auto)

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