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
  K is having_valuation implies vp(v) is prime
  proof
    assume
A1: K is having_valuation;
    hence vp(v) is proper by Th76;
    let a, b be Element of ValuatRing v such that
A2: a*b in vp(v);
    assume not a in vp(v);
    then
A3: v.a = 0 by A1,Th77;
    assume
A4: not b in vp(v);
    reconsider x = a, y = b as Element of K by A1,Th51;
A5: a*b = x*y by A1,Th55;
A6: v.y = 0 by A1,A4,Th77;
    v.(x*y) = v.x+v.y by A1,Def8
    .= 0 by A3,A6;
    hence thesis by A1,A2,A5,Th61;
  end;
