reserve k,n,n1,m,m1,m0,h,i,j for Nat,
  a,x,y,X,X1,X2,X3,X4,Y for set;
reserve L,L1,L2 for FinSequence;
reserve F,F1,G,G1,H for LTL-formula;
reserve W,W1,W2 for Subset of Subformulae H;
reserve v for LTL-formula;
reserve N,N1,N2,N10,N20,M for strict LTLnode over v;
reserve w for Element of Inf_seq(AtomicFamily);
reserve R1,R2 for Real_Sequence;
reserve s,s0,s1,s2 for elementary strict LTLnode over v;
reserve q for sequence of LTLStates(v);
reserve U for Choice_Function of BOOL Subformulae v;

theorem Th64:
  H is Sub_atomic iff H is atomic or H is negative &
  the_argument_of H is atomic
proof
  thus H is Sub_atomic implies H is atomic or H is negative & the_argument_of
  H is atomic
  proof
    assume
A1: H is Sub_atomic;
    per cases by A1;
    suppose
      H is atomic;
      hence thesis;
    end;
    suppose
A2:   ex G being LTL-formula st G is atomic & H = 'not' G;
      then H is negative;
      hence thesis by A2,MODELC_2:def 18;
    end;
  end;
  thus H is atomic or H is negative & the_argument_of H is atomic implies H is
  Sub_atomic
  proof
    assume
A3: H is atomic or H is negative & the_argument_of H is atomic;
    per cases by A3;
    suppose
      H is atomic;
      hence thesis;
    end;
    suppose
A4:   H is negative & the_argument_of H is atomic;
      then H = 'not' the_argument_of H by MODELC_2:def 18;
      hence thesis by A4;
    end;
  end;
end;
