reserve x,y for set;

theorem Th46:
  for I, J being set for A being ManySortedSet of [:I,I:], B being
  ManySortedSet of [:J,J:] st A cc= B holds ~A cc= ~B
proof
  let I, J be set;
  let A be ManySortedSet of [:I,I:], B be ManySortedSet of [:J,J:] such that
A1: [:I,I:] c= [:J,J:] and
A2: for x st x in [:I,I:] holds A.x c= B.x;
  thus [:I,I:] c= [:J,J:] by A1;
  let x;
  assume x in [:I,I:];
  then consider x1,x2 being object such that
A3: x1 in I & x2 in I and
A4: x = [x1,x2] by ZFMISC_1:def 2;
A5: [x2,x1] in [:I,I:] by A3,ZFMISC_1:def 2;
  dom A = [:I,I:] by PARTFUN1:def 2;
  then
A6: ~A.(x1,x2) = A.(x2,x1) by A5,FUNCT_4:def 2;
  dom B = [:J,J:] by PARTFUN1:def 2;
  then ~B.(x1,x2) = B.(x2,x1) by A1,A5,FUNCT_4:def 2;
  hence thesis by A2,A4,A5,A6;
end;
