reserve D,D9 for non empty set;
reserve R for Ring;
reserve G,H,S for non empty ModuleStr over R;
reserve UN for Universe;
reserve R for Ring;
reserve G, H for LeftMod of R;
reserve G1, G2, G3 for LeftMod of R;
reserve f for LModMorphismStr over R;
reserve a,b,c for Element of {0,1,2};

theorem Th26:
  for F being non empty doubleLoopStr st for x,y,z being Scalar of
F holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.F) = x & x+(-x) = (0.F) & x*y = y*
  x & (x*y)*z = x*(y*z) & 1.F*x = x & x*(1.F) = x & (x<>(0.F) implies ex y be
  Scalar of F st x*y = 1.F) & 0.F <> 1.F & x*(y+z) = x*y+x*z holds F is Field
proof
  let F be non empty doubleLoopStr such that
A1: for x,y,z being Scalar of F holds x+y = y+x & (x+y)+z = x+(y+z) & x+
(0.F) = x & x+(-x) = (0.F) & x*y = y*x & (x*y)*z = x*(y*z) & (1.F)*x = x & x*(
1.F) = x & (x<>(0.F) implies ex y be Scalar of F st x*y = 1.F) & 0.F <> 1.F & x
  *(y+z) = x*y+x*z;
A2: for x being Scalar of F st x<>0.F ex y be Scalar of F st y*x = 1.F
  proof
    let x being Scalar of F;
    assume x<>0.F;
    then consider y be Scalar of F such that
A3: x*y = 1.F by A1;
    take y;
    thus thesis by A1,A3;
  end;
A4: now
    let x,y,z be Scalar of F;
    thus x+y = y+x & (x+y)+z = x+(y+z) & x+(0.F) = x & x+(-x) = (0.F) & x*y =
y*x & (x*y)*z = x*(y*z) & (1.F)*x = x & x*(1.F) = x & (x<>(0.F) implies ex y be
    Scalar of F st x*y = 1.F) & 0.F <> 1.F & x*(y+z) = x*y+x*z by A1;
    thus (y+z)*x = x*(y+z) by A1
      .= x*y + x*z by A1
      .= y*x + x*z by A1
      .= y*x + z*x by A1;
  end;
  F is right_complementable
  proof
    let v be Element of F;
    take -v;
    thus thesis by A4;
  end;
  hence thesis by A2,A4,GROUP_1:def 3,def 12,RLVECT_1:def 2,def 3,def 4
,STRUCT_0:def 8,VECTSP_1:def 6,def 7,def 9;
end;
