
theorem
  for A1,A2 be non empty set, B be set, D1 be thin_cylinder of A1,B st
  A1 c= A2 ex D2 be thin_cylinder of A2,B st D1 c= D2
proof
  let A1,A2 be non empty set, B be set, D1 be thin_cylinder of A1,B;
  consider Bo being Subset of B,yo1 being Function of Bo,A1 such that
A1: Bo is finite and
A2: D1 = { y where y is Function of B,A1: y|Bo = yo1 } by Th1;
  assume
A3: A1 c= A2;
  then reconsider yo2=yo1 as Function of Bo,A2 by FUNCT_2:7;
  set D2= { y where y is Function of B,A2: y|Bo = yo2 };
A4: now
    let x be object;
    assume x in D1;
    then consider y1 be Function of B,A1 such that
A5: x=y1 and
A6: y1|Bo = yo1 by A2;
    reconsider y2=y1 as Function of B,A2 by A3,FUNCT_2:7;
    y2|Bo = yo1 by A6;
    hence x in D2 by A5;
  end;
  D2= cylinder0(A2,B,Bo,yo2) by Def1;
  then reconsider D2 as thin_cylinder of A2,B by A1,Def2;
  take D2;
  thus thesis by A4;
end;
