theorem Th8:
  p = P!ll & r = J.P implies (not v*'ll in r iff Valid(p,J).v = FALSE)
proof
  assume that
A1: p = P!ll and
A2: r = J.P;
A3: now
    assume Valid(p,J).v = FALSE;
    then (ll 'in' (J.P)).v <> TRUE by A1,Lm1;
    hence not v*'ll in r by A2,Def4;
  end;
  now
    assume not v*'ll in r;
    then (ll 'in' (J.P)).v = FALSE by A2,Def4;
    hence Valid(p,J).v = FALSE by A1,Lm1;
  end;
  hence thesis by A3;
end;
