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 Th40: p => r => (p => s => (p => (r '&&' s))) is ctaut
  proof
    let g;
    set v = VAL g;
A1: v.p = 1 or v.p = 0 by XBOOLEAN:def 3;
A2: v.r = 1 or v.r = 0 by XBOOLEAN:def 3;
A3: v.(p => s => (p => (r '&&' s)))
    = v.(p =>s) => v.(p => (r '&&' s)) by LTLAXIO1:def 15
    .= v.p => v.s => v.(p => (r '&&' s)) by LTLAXIO1:def 15
    .= v.p => v.s => (v.p => v.(r '&&' s)) by LTLAXIO1:def 15
    .= v.p => v.s => (v.p => (v.r '&' v.s)) by LTLAXIO1:31;
A4: v.s = 1 or v.s = 0 by XBOOLEAN:def 3;
    v.(p => r) = v.p => v.r by LTLAXIO1:def 15;
   hence v.(p => r => (p => s => (p => (r '&&' s))))
   = v.p => v.r => (v.p => v.s => (v.p => (v.r '&' v.s))) by LTLAXIO1:def 15,A3
   .= 1 by A1,A2,A4;
 end;
