reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;
reserve L for non empty ZeroStr;
reserve x for Element of L;
reserve p,q for Polynomial of F_Real;

theorem Th47:
  poly_diff(p+q) = poly_diff(p) + poly_diff(q)
  proof
    let n be Element of NAT;
A1: (poly_diff(p)).n = p.(n+1) * (n+1) by Def5;
A2: (poly_diff(q)).n = q.(n+1) * (n+1) by Def5;
A3: (p+q).(n+1) = p.(n+1) + q.(n+1) by NORMSP_1:def 2;
    thus (poly_diff(p+q)).n = (p+q).(n+1) * (n+1) by Def5
    .= (poly_diff(p)).n + (poly_diff(q)).n by A1,A2,A3
    .= (poly_diff(p) + poly_diff(q)).n by NORMSP_1:def 2;
  end;
