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
  r => untn(p,q) => (r => (('not' p) '&&' ('not' q)) => ('not' r)) is ctaut
  proof
    let g;
    set v = VAL g,pq = p 'U' q,np='not'p,nq='not'q,nr='not'r;
A1: v.tf = 0 by LTLAXIO1:def 15;
A2: v.p = 1 or v.p = 0 by XBOOLEAN:def 3;
A3: v.(r => (np '&&' nq) => nr)
    = v.(r => (np '&&' nq)) => v.nr by LTLAXIO1:def 15
    .= v.r => v.(np '&&' nq) => v.nr by LTLAXIO1:def 15
    .= v.r => (v.np '&' v.nq) => v.nr by LTLAXIO1:31
    .= v.r => ((v.p => v.tf) '&' v.nq) => v.nr by LTLAXIO1:def 15
    .= v.r => ((v.p => v.tf) '&' (v.q => v.tf)) => v.nr by LTLAXIO1:def 15
    .= v.r => ((v.p => v.tf) '&' (v.q => v.tf)) => (v.r => v.tf)
    by LTLAXIO1:def 15;
A4: v.q = 1 or v.q = 0 by XBOOLEAN:def 3;
A5: v.r = 1 or v.r = 0 by XBOOLEAN:def 3;
    v.(r => untn(p,q)) = v.r => v.untn(p,q) by LTLAXIO1:def 15
    .= v.r => (v.q 'or' v.(p '&&' pq)) by Th5
    .= v.r => (v.q 'or' (v.p '&' v.pq)) by LTLAXIO1:31
    .= v.r => (v.q 'or' (v.p '&' g.pq)) by LTLAXIO1:def 15;
   hence v.(r => untn(p,q) => (r => (np '&&' nq) => nr))
   = v.r => (v.q 'or' (v.p '&' g.pq)) => (v.r => ((v.p => v.tf) '&'
   (v.q => v.tf)) => (v.r => v.tf)) by LTLAXIO1:def 15,A3
   .= 1 by A2,A5,A4,A1;
 end;
