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);

theorem Th54:
  H is Until & H in the LTLold of CastNode(q.1,v) & (for i holds
CastNode(q.(i+1),v) is_next_of CastNode(q.i,v)) implies (for i st i>=1 holds (H
  in the LTLold of CastNode(q.i,v)) & (the_left_argument_of H in the LTLold of
CastNode(q.i,v)) & not (the_right_argument_of H in the LTLold of CastNode(q.i,v
))) or ex j st j>=1 & the_right_argument_of H in the LTLold of CastNode(q.j,v)
  & for i st 1<=i & i<j holds H in the LTLold of CastNode(q.i,v) &
  the_left_argument_of H in the LTLold of CastNode(q.i,v)
proof
  deffunc Node(Nat) = CastNode(q.$1,v);
  assume that
A1: H is Until and
A2: H in the LTLold of Node(1) & for i holds Node(i+1) is_next_of Node(i );
  set G = the_right_argument_of H;
  set F = the_left_argument_of H;
  H = F 'U' G by A1,MODELC_2:8;
  hence thesis by A2,Lm32;
end;
