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 Th53: X |- p => q & X |- r => s implies X |- (p '&&' r) => (q '&&' s)
  proof
   assume that
A1: X |- p => q and
A2: X |- r => s;
    (p => q) => ((r => s) => ((p '&&' r) => (q '&&' s))) is ctaut by Th45;
    then (p => q) => ((r => s) => ((p '&&' r) => (q '&&' s)))
    in LTL_axioms by LTLAXIO1:def 17;
    then
    X |- (p => q) => ((r => s) => ((p '&&' r) => (q '&&' s))) by LTLAXIO1:42;
    then X |- (r => s) => ((p '&&' r) => (q '&&' s)) by LTLAXIO1:43,A1;
    hence X |- (p '&&' r) => (q '&&' s) by LTLAXIO1:43,A2;
  end;
