reserve k,n,m for Nat,
  a,x,X,Y for set,
  D,D1,D2,S for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for LTL-formula;
reserve sq,sq9 for FinSequence;

theorem Th17:
  H is_immediate_constituent_of F 'U' G iff H = F or H =G
proof
  thus H is_immediate_constituent_of F 'U' G implies H = F or H =G
  proof
    set Z= F 'U' G;
A1: now
      assume
A2:   Z = 'not' H or Z = 'X' H;
      Z.1 = 4 by Th12;
      hence contradiction by A2,Th12;
    end;
A3: now
      given H1 such that
A4:   Z = H '&' H1 or Z = H1 '&' H or Z = H 'or' H1 or Z = H1 'or' H
      or Z = H 'R' H1 or Z = H1 'R' H;
      Z.1 = 4 by Th12;
      hence contradiction by A4,Th12;
    end;
    assume H is_immediate_constituent_of F 'U' G;
    then
    Z = 'not' H or Z = 'X' H or ex H1 st Z = H '&' H1 or Z = H1 '&' H or Z
= H 'or' H1 or Z = H1 'or' H or Z = H 'U' H1 or Z = H1 'U' H or Z = H 'R' H1 or
    Z = H1 'R' H;
    hence thesis by A1,A3,Lm14;
  end;
  thus thesis;
end;
