theorem Th24:
  S1`2 = S2`2 implies (J,v.Val_S(v,S1) |= S1 & J,v.Val_S(v,S2) |=
  S2 iff J,v.Val_S(v,CQCSub_&(S1,S2)) |= CQCSub_&(S1,S2))
proof
  assume
A1: S1`2 = S2`2;
  then Val_S(v,S1) = Val_S(v,CQCSub_&(S1,S2)) by Th21;
  then
A2: J,v.Val_S(v,S1) |= S1`1 & J,v.Val_S(v,S1) |= S2`1 iff J,v.Val_S(v,
  CQCSub_&(S1,S2)) |= (S1`1) '&' (S2`1) by VALUAT_1:18;
  J,v.Val_S(v,CQCSub_&(S1,S2)) |= (S1`1) '&' (S2`1) iff J,v.Val_S(v,
  CQCSub_&(S1,S2)) |= CQCSub_&(S1,S2)`1 by A1,Th20;
  hence thesis by A1,A2;
end;
