reserve x,y,z,x1,x2,x3,x4,y1,y2,s for Variable,
  M for non empty set,
  a,b for set,
  i,j,k for Element of NAT,
  m,m1,m2,m3,m4 for Element of M,
  H,H1,H2 for ZF-formula,
  v,v9,v1,v2 for Function of VAR,M;

theorem Th6:
  x <> y & y <> z & z <> x implies v/(x,m1)/(y,m2)/(z,m3) = v/(z,m3
  )/(y,m2)/(x,m1) & v/(x,m1)/(y,m2)/(z,m3) = v/(y,m2)/(z,m3)/(x,m1)
proof
  assume that
A1: x <> y and
A2: y <> z and
A3: z <> x;
A4: v/(z,m3)/(y,m2) = v/(y,m2)/(z,m3) by A2,FUNCT_7:33;
  v/(x,m1)/(y,m2) = v/(y,m2)/(x,m1) by A1,FUNCT_7:33;
  hence thesis by A3,A4,FUNCT_7:33;
end;
