reserve T, X, Y for Subset of MC-wff;
reserve p, q, r, s for Element of MC-wff;

theorem Th13:
  X c= Y implies CnIPC(X) c= CnIPC(Y)
proof
  assume
A1: X c= Y;
  thus CnIPC(X) c= CnIPC(Y)
  proof
    let a be object;
    assume
A2: a in CnIPC(X);
    then reconsider t = a as Element of MC-wff;
    for T st T is IPC_theory & Y c= T holds t in T
    proof
      let T such that
A3:   T is IPC_theory and
A4:   Y c= T;
      X c= T by A1,A4;
      hence thesis by A2,A3,Def15;
    end;
    hence thesis by Def15;
  end;
end;
