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 Th27:
  Z_3 is Fanoian Field
proof
  set F = Z_3;
  reconsider a = 0.F, b = 1.F as Element of {0,1,2};
  now
    let x,y,z be Scalar of Z_3;
    thus x+y = y+x & (x+y)+z = x+(y+z) & x+(0.Z_3) = x &
x+(-x) = (0.Z_3) & x*y
= y*x & (x*y)*z = x*(y*z) & (1.Z_3)*x = x & x*(1.Z_3) = x &
  (x<>0.Z_3 implies ex y
    be Scalar of Z_3 st x*y = 1.Z_3) & 0.Z_3 <> 1.Z_3 & x*(y+z) = x*y+x*z
    proof
      reconsider X=x, Y=y, Z=z as Element of {0,1,2};
A1:   x*y = X*Y & x*z = X*Z by Th23;
      thus x+y = X+Y by Th23
        .= Y+X by Th25
        .= y+x by Th23;
      thus (x+y)+z = (X+Y)+Z by Th24
        .= X+(Y+Z) by Th25
        .= x+(y+z) by Th24;
      thus x+(0.Z_3) = X+a by Th23
        .= x by Th25;
      -x = -X by Th23;
      hence x+(-x) = X+(-X) by Th23
        .= (0.Z_3) by Th25;
      thus x*y = X*Y by Th23
        .= Y*X by Th25
        .= y*x by Th23;
      thus (x*y)*z = (X*Y)*Z by Th24
        .= X*(Y*Z) by Th25
        .= x*(y*z) by Th24;
      thus (1.Z_3)*x = b*X by Th23
        .= x by Th25;
      thus x*(1.Z_3) = X*b by Th23
        .= b*X by Th25
        .= x by Th25;
      thus x <> 0.Z_3 implies ex y being Scalar of Z_3 st x*y = 1.Z_3
      proof
        assume x <> 0.Z_3;
        then consider Y being Element of {0,1,2} such that
A2:     X*Y = b by Th25;
        reconsider y=Y as Scalar of Z_3;
        take y;

        thus thesis by A2,Th23;
      end;
      thus 0.Z_3 <> 1.Z_3;
      y+z = Y+Z by Th23;
      hence x*(y+z) = X*(Y+Z) by Th23
        .= X*Y + X*Z by Th25
        .= x*y+x*z by A1,Th23;
    end;
  end;
  then reconsider F as Field by Th26;
  1.F + 1.F = b + b by Def15
    .= 2 by Def13;
  hence thesis by Th22,VECTSP_1:def 19;
end;
