reserve A,B,C for non empty set,
  f for Function of [:A,B:],C;
reserve K for non empty doubleLoopStr;
reserve V for non empty ModuleStr over K;
reserve W for non empty RightModStr over K;

theorem Th19:
  for K being Ring, V being LeftMod of K holds opp V is strict
  RightMod of opp K
proof
  let K be Ring, V be LeftMod of K;
  set R=opp(K);
  reconsider W=opp(V) as non empty RightModStr over R;
A1: the addLoopStr of opp V = the addLoopStr of V by Th7;
  then
A2: for a,b be Element of opp V for x,y be Element of V st x = a & b = y
  holds a + b = x + y;
A3: opp V is Abelian add-associative right_zeroed right_complementable
  proof
    thus opp V is Abelian
    proof
      let a,b be Element of opp V;
      reconsider x = a, y = b as Element of V by Th7;
      thus a + b = y + x by A2
        .= b + a by A1;
    end;
    hereby
      let a,b,c be Element of opp V;
      reconsider x = a, y = b, z = c as Element of V by Th7;
      thus a + b + c = x + y + z by A1
        .= x + (y + z) by RLVECT_1:def 3
        .= a + (b + c) by A1;
    end;
    hereby
      let a be Element of opp V;
      reconsider x = a as Element of V by Th7;
      thus a + 0.opp V = x + 0.V by A1
        .= a by RLVECT_1:4;
    end;
    let a be Element of opp V;
    reconsider x = a as Element of V by Th7;
    consider b being Element of V such that
A4: x + b = 0.V by ALGSTR_0:def 11;
    reconsider b9 = b as Element of opp V by Th7;
    take b9;
    thus thesis by A1,A4;
  end;
  now
    let x,y be Scalar of R, v,w be Vector of W;
    reconsider p=v,q=w as Vector of V by Th7;
    reconsider a=x,b=y as Scalar of K;
A5: b*p=v*y by Th12;
A6: a*q=w*x by Th12;
A7: a*p=v*x by Th12;
    v+w=p+q by Th13;
    hence (v+w)*x = a*(p+q) by Th12
      .= a*p+a*q by VECTSP_1:def 14
      .= v*x+w*x by A7,A6,Th13;
    thus v*(x+y) = (a+b)*p by Th12
      .= a*p+b*p by VECTSP_1:def 15
      .= v*x+v*y by A5,A7,Th13;
    thus v*(y*x) = (a*b)*p by Lm3,Th12
      .= a*(b*p) by VECTSP_1:def 16
      .= (v*y)*x by A5,Th12;
    thus v*(1_R) = (1_K)*p by Th12
      .= v;
  end;
  hence thesis by A3,VECTSP_2:def 9;
end;
