theorem Th54:
  K is having_valuation implies
  for x, y being Element of K,
  x1, y1 being Element of ValuatRing v st x = x1 & y = y1 holds
  x + y = x1 + y1
  proof
    set R = ValuatRing v;
    set c = NonNegElements v;
    assume
A1: K is having_valuation;
    let x, y be Element of K,
    x1, y1 be Element of R such that
A2: x = x1 & y = y1;
A3: c = the carrier of R by A1,Def12;
A4: the addF of R = (the addF of K) | [:c,c:] by A1,Def12;
    thus x1+y1 = (the addF of R).[x1,y1]
    .= x+y by A3,A4,A2,FUNCT_1:49;
  end;
