reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th39:
  for p be Element of TOP-REAL n,
      f be Element of REAL-NS n,
      H be Subset of TOP-REAL n,
      I be Subset of REAL-NS n
    st p = f & H = I
  holds p + H = f + I
  proof
    let p be Element of TOP-REAL n,
        f be Element of REAL-NS n,
        H be Subset of TOP-REAL n,
        I be Subset of REAL-NS n;

    assume
    A1: p = f & H = I;
    for x be object holds
    x in p+H iff x in f+I
    proof
      let x be object;
      hereby
        assume x in p+H;
        then
        consider v be Element of TOP-REAL n such that
        A2: x = p+v & v in H;
        reconsider w = v as Element of REAL-NS n by Th4;
        x = f+w by A1,A2,Th7;
        hence x in f+I by A2,A1;
      end;
      assume x in f+I;
      then consider v be Element of REAL-NS n such that
      A3: x = f+v & v in I;

      reconsider w = v as Element of TOP-REAL n by Th4;
      x = p+w by A1,A3,Th7;
      hence x in p+H by A1,A3;
    end;
    hence thesis by TARSKI:2;
  end;
