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' (p '&&' s) => 'not' ((r '&&' s) '&&' (p '&&' q)) 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' ((r '&&' s) '&&' (p '&&' q)))
    = v.((r '&&' s) '&&' (p '&&' q)) => v.tf by LTLAXIO1:def 15
    .= v.(r '&&' s) '&' v.(p '&&' q) => v.tf by LTLAXIO1:31
    .= v.r '&' v.s '&' v.(p '&&' q) => v.tf by LTLAXIO1:31
    .= v.r '&' v.s '&' (v.p '&' v.q) => v.tf by LTLAXIO1:31;
A4: v.s = 1 or v.s = 0 by XBOOLEAN:def 3;
    v.('not' (p '&&' s)) = v.(p '&&' s) => v.tf by LTLAXIO1:def 15
    .= v.p '&' v.s => v.tf by LTLAXIO1:31;
   hence v.('not' (p '&&' s) => 'not' ((r '&&' s) '&&' (p '&&' q)))
   = v.p '&' v.s => v.tf => (v.r '&' v.s '&' (v.p '&' v.q) => v.tf)
   by LTLAXIO1:def 15,A3
   .= 1 by A2,A4,A1;
 end;
