reserve x, y, z, s for ExtReal;
reserve i, j for Integer;
reserve n, m for Nat;
reserve x, y, v, u for ExtInt;
reserve
  D for non empty doubleLoopStr,
  A for Subset of D;
reserve K for Field-like non degenerated
  associative add-associative right_zeroed right_complementable
  distributive Abelian non empty doubleLoopStr,
  a, b, c for Element of K;
reserve v for Valuation of K;

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;
