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 Th8:
  v/(x1,m1)/(x2,m2)/(x1,m) = v/(x2,m2)/(x1,m) & v/(x1,m1)/(x2,m2)/(
x3,m3)/(x1,m) = v/(x2,m2)/(x3,m3)/(x1,m) & v/(x1,m1)/(x2,m2)/(x3,m3)/(x4,m4)/(
  x1,m) = v/(x2,m2)/(x3,m3)/(x4,m4)/(x1,m)
proof
A1: v/(x1,m1)/(x2,m2)/(x1,m) = v/(x2,m2)/(x1,m1)/(x1,m) or x1 = x2 by
FUNCT_7:33;
  hence v/(x1,m1)/(x2,m2)/(x1,m) = v/(x2,m2)/(x1,m) by FUNCT_7:34;
  x1 = x3 or x1 <> x3;
  then
A2: v/(x1,m1)/(x2,m2)/(x3,m3)/(x1,m) = v/(x1,m1)/(x2,m2)/(x1,m)/(x3,m3 ) & v
/(x2,m2)/(x3,m3)/(x1,m) = v/(x2,m2)/(x1,m)/(x3,m3) or v/(x1,m1)/(x2,m2)/(x3,m3)
/(x1,m) = v/(x1,m1)/(x2,m2)/(x1,m) & v/(x2,m2)/(x3,m3)/(x1,m) = v/(x2,m2)/(x1,m
  ) by FUNCT_7:33,34;
  hence v/(x1,m1)/(x2,m2)/(x3,m3)/(x1,m) = v/(x2,m2)/(x3,m3)/(x1,m) by A1,
FUNCT_7:34;
  x1 = x4 or x1 <> x4;
  then
  v/(x1,m1)/(x2,m2)/(x3,m3)/(x4,m4)/(x1,m) = v/(x1,m1)/(x2,m2)/(x3,m3)/(x1
,m)/(x4,m4) & v/(x2,m2)/(x3,m3)/(x1,m)/(x4,m4) = v/(x2,m2)/(x3,m3)/(x4,m4)/(x1,
m) or v/(x1,m1)/(x2,m2)/(x3,m3)/(x4,m4)/(x1,m) = v/(x1,m1)/(x2,m2)/(x3,m3)/(x1,
  m) & v/(x2,m2)/(x3,m3)/(x4,m4)/(x1,m) = v/(x2,m2)/(x3,m3)/(x1,m) by
FUNCT_7:33,34;
  hence thesis by A1,A2,FUNCT_7:34;
end;
