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 Th55:
  Eval(p+q) = Eval(p) + Eval(q)
  proof
    let r be Element of REAL;
    set s = In(r,F);
    (Eval(p)).r = eval(p,s) & (Eval(q)).r = eval(q,s) by POLYNOM5:def 13;
    hence (Eval(p) + Eval(q)).r = eval(p,s) + eval(q,s) by VALUED_1:1
    .= eval(p+q,s) by POLYNOM4:19
    .= (Eval(p+q)).r by POLYNOM5:def 13;
  end;
