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 X |- p => q & X |- p => r implies X |- p => (q '&&' r)
  proof
    assume that
A1: X |- p => q and
A2: X |- p => r;
    set qr = q '&&' r;
    p => q => (p => r => (p => qr)) is ctaut by Th40;then
    p => q => (p => r => (p => qr)) in LTL_axioms by LTLAXIO1:def 17;
    then X |- p => q => (p => r => (p => qr)) by LTLAXIO1:42;
    then X |- p => r => (p => qr) by LTLAXIO1:43,A1;
    hence X |- p => qr by LTLAXIO1:43,A2;
  end;
