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 Th7:
  x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4
implies v/(x1,m1)/(x2,m2)/(x3,m3)/(x4,m4) = v/(x2,m2)/(x3,m3)/(x4,m4)/(x1,m1) &
v/(x1,m1)/(x2,m2)/(x3,m3)/(x4,m4) = v/(x3,m3)/(x4,m4)/(x1,m1)/(x2,m2) & v/(x1,
  m1)/(x2,m2)/(x3,m3)/(x4,m4) = v/(x4,m4)/(x2,m2)/(x3,m3)/(x1,m1)
proof
  assume that
A1: x1 <> x2 and
A2: x1 <> x3 and
A3: x1 <> x4 and
A4: x2 <> x3 and
A5: x2 <> x4 and
A6: x3 <> x4;
A7: v/(x1,m1)/(x3,m3)/(x4,m4)/(x2,m2) = v/(x3,m3)/(x4,m4)/(x1,m1)/(x2,m2) by A2
,A3,A6,Th6;
A8: v/(x2,m2)/(x3,m3)/(x1,m1)/(x4,m4) = v/(x2,m2)/(x3,m3)/(x4,m4)/(x1,m1) by A3
,FUNCT_7:33;
  v/(x1,m1)/(x2,m2)/(x3,m3) = v/(x2,m2)/(x3,m3)/(x1,m1) by A1,A2,A4,Th6;
  hence thesis by A4,A5,A6,A8,A7,Th6;
end;
