
theorem Th3:
  for A be non trivial set, B be set, Bo1 be set, yo1 being
Function of Bo1,A, Bo2 be set, yo2 being Function of Bo2,A st Bo1 c= B & Bo2 c=
  B & cylinder0(A,B,Bo1,yo1) = cylinder0(A,B,Bo2,yo2) holds Bo1=Bo2 & yo1=yo2
proof
  let A be non trivial set, B be set, Bo1 be set,yo1 being Function of Bo1,A,
  Bo2 be set,yo2 being Function of Bo2,A;
  assume that
A1: Bo1 c= B and
A2: Bo2 c= B and
A3: cylinder0(A,B,Bo1,yo1) = cylinder0(A,B,Bo2,yo2);
A4: { y where y is Function of B,A : y|Bo1 = yo1 } = cylinder0(A,B,Bo1,yo1)
  by A1,Def1;
  then consider y0 be object such that
A5: y0 in { y where y is Function of B,A : y|Bo1 = yo1 } by XBOOLE_0:def 1;
A6: ex y be Function of B,A st y0=y & y|Bo1 = yo1 by A5;
A7: { y where y is Function of B,A : y|Bo2 = yo2 } = cylinder0(A,B,Bo2,yo2)
  by A2,Def1;
  hence Bo1=Bo2 by A1,A2,A3,A4,Lm4;
  ex w be Function of B,A st y0=w & w|Bo2 = yo2 by A3,A4,A7,A5;
  hence thesis by A1,A2,A3,A4,A7,A6,Lm4;
end;
