theorem Th25:
  S1`2 = S2`2 & (for v holds (J,v |= CQC_Sub(S1) iff J,v.Val_S(v,
  S1) |= S1)) & (for v holds (J,v |= CQC_Sub(S2) iff J,v.Val_S(v,S2) |= S2))
implies for v holds (J,v |= CQC_Sub(CQCSub_&(S1,S2)) iff J,v.Val_S(v,CQCSub_&(
  S1,S2)) |= CQCSub_&(S1,S2))
proof
  assume that
A1: S1`2 = S2`2 and
A2: ( for v holds (J,v |= CQC_Sub(S1) iff J,v.Val_S(v,S1) |= S1))& for v
  holds (J, v |= CQC_Sub(S2) iff J,v.Val_S(v,S2) |= S2);
  let v;
A3: J,v |= (CQC_Sub(S1)) & J,v |= (CQC_Sub(S2)) iff J,v.Val_S(v,S1) |= S1 &
  J,v.Val_S(v,S2) |= S2 by A2;
  J,v |= CQC_Sub(CQCSub_&(S1,S2)) iff J,v |= (CQC_Sub(S1)) '&' (CQC_Sub(S2
  )) by A1,Th23;
  hence thesis by A1,A3,Th24,VALUAT_1:18;
end;
