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 Th20:
  for K being Ring, W being RightMod of K holds
    opp W is strict LeftMod of opp K
proof
  let K be Ring, W be RightMod of K;
  set R=opp(K);
  reconsider V=opp(W) as non empty ModuleStr over R;
A1: the addLoopStr of opp W = the addLoopStr of W by Th9;
  then
A2: for a,b be Element of opp W for x,y be Element of W st x = a & b = y
  holds a + b = x + y;
A3: opp W is Abelian add-associative right_zeroed right_complementable
  proof
    thus opp W is Abelian
    proof
      let a,b be Element of opp W;
      reconsider x = a, y = b as Element of W by Th9;
      thus a + b = y + x by A2
        .= b + a by A1;
    end;
    hereby
      let a,b,c be Element of opp W;
      reconsider x = a, y = b, z = c as Element of W by Th9;
      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 W;
      reconsider x = a as Element of W by Th9;
      thus a + 0.opp W = x + 0.W by A1
        .= a by RLVECT_1:4;
    end;
    let a be Element of opp W;
    reconsider x = a as Element of W by Th9;
    consider b being Element of W such that
A4: x + b = 0.W by ALGSTR_0:def 11;
    reconsider b9 = b as Element of opp W by Th9;
    take b9;
    thus thesis by A1,A4;
  end;
  now
    let x,y be Scalar of R, v,w be Vector of V;
    reconsider p=v,q=w as Vector of W by Th9;
    reconsider a=x,b=y as Scalar of K;
A5: p*b=y*v by Th15;
A6: q*a=x*w by Th15;
A7: p*a=x*v by Th15;
    v+w=p+q by Th16;
    hence x*(v+w) = (p+q)*a by Th15
      .= p*a+q*a by VECTSP_2:def 9
      .= x*v+x*w by A7,A6,Th16;
    thus (x+y)*v = p*(a+b) by Th15
      .= p*a+p*b by VECTSP_2:def 9
      .= x*v+y*v by A5,A7,Th16;
    x*y=b*a by Lm3;
    hence (x*y)*v = p*(b*a) by Th15
      .= (p*b)*a by VECTSP_2:def 9
      .= x*(y*v) by A5,Th15;
    thus (1_R)*v = p*(1_K) by Th15
      .= v by VECTSP_2:def 9;
  end;
  hence thesis by A3,VECTSP_1:def 14,def 15,def 16,def 17;
end;
