reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;
reserve X,Y,Z for non empty set;

theorem Th55:
  for X being set for D being Subset-Family of X st union D = X
for A being Subset of D, B being Subset of X st B = union A holds
  B` c= union A`
proof
  let X be set;
  let D be Subset-Family of X such that
A1: union D = X;
  let A be Subset of D, B be Subset of X such that
A2: B = union A;
  let e be object;
  assume
A3: e in B`;
  then consider u being set such that
A4: e in u and
A5: u in D by A1,TARSKI:def 4;
  not e in B by A3,XBOOLE_0:def 5;
  then not u in A by A2,A4,TARSKI:def 4;
  then u in A` by A5,SUBSET_1:29;
  hence thesis by A4,TARSKI:def 4;
end;
