reserve A,B,p,q,r,s for Element of LTLB_WFF,
  i,j,k,n for Element of NAT,
  X for Subset of LTLB_WFF,
  f,f1 for FinSequence of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN;

theorem 'not' (s '&&' q) => (p => q => (p => 'not' s)) is ctaut
  proof
    let g;
    set v = VAL g;
A1: v.tf = 0 by LTLAXIO1:def 15;
A2: v.p = 1 or v.p = 0 by XBOOLEAN:def 3;
A3: v.('not' (s '&&' q)) = v.(s '&&' q) => v.tf by LTLAXIO1:def 15
    .= v.s '&' v.q => v.tf by LTLAXIO1:31;
A4: v.s = 1 or v.s = 0 by XBOOLEAN:def 3;
A5: v.q = 1 or v.q = 0 by XBOOLEAN:def 3;
    v.(p => q => (p => 'not' s)) = v.(p =>q) => v.(p => 'not' s)
    by LTLAXIO1:def 15
    .= v.p => v.q => v.(p => 'not' s) by LTLAXIO1:def 15
    .= v.p => v.q => (v.p => v.('not' s)) by LTLAXIO1:def 15
    .= v.p => v.q => (v.p => (v.s => v.tf)) by LTLAXIO1:def 15;
   hence v.('not' (s '&&' q) => (p => q => (p => 'not' s)))
   = v.s '&' v.q => v.tf => (v.p => v.q => (v.p => (v.s => v.tf)))
   by LTLAXIO1:def 15,A3
   .= 1 by A2,A5,A4,A1;
 end;
