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;

theorem Th43:
  for X being non empty set, S being a_partition of X,
      A being Subset of S holds (union S) \ (union A) = union (S \ A)
proof
  let X be non empty set;
  let S be a_partition of X;
  let A be Subset of S;
  thus (union S) \ (union A) c= union (S \ A)
  proof
    let y be object;
    assume
A1: y in (union S) \ (union A);
    then y in (union S) by XBOOLE_0:def 5;
    then consider z being set such that
A2: y in z and
A3: z in S by TARSKI:def 4;
    not y in (union A) by A1,XBOOLE_0:def 5;
    then not z in A by A2,TARSKI:def 4;
    then z in S \ A by A3,XBOOLE_0:def 5;
    hence thesis by A2,TARSKI:def 4;
  end;
  thus union (S \ A) c= (union S) \ (union A)
  proof
    let y be object;
    assume y in union(S \ A);
    then consider z being set such that
A4: y in z and
A5: z in S \ A by TARSKI:def 4;
A6: z in S by A5,XBOOLE_0:def 5;
A7: not z in A by A5,XBOOLE_0:def 5;
A8: now
      let w be set;
      assume
A9:   w in A;
      then w in S;
      then z misses w by A6,A7,A9,Def4;
      hence not y in w by A4,XBOOLE_0:3;
    end;
A10: now
      assume y in union A;
      then ex v being set st y in v & v in A by TARSKI:def 4;
      hence contradiction by A8;
    end;
    y in union S by A4,A6,TARSKI:def 4;
    hence thesis by A10,XBOOLE_0:def 5;
  end;
end;
