reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;
reserve A for set,
  x,y,z for Element of Fin A;
reserve X,Y for non empty set,
  A for set,
  f for (Function of X, Fin A),
  i,j,k for (Element of X);

theorem Th54:
  for B being Element of Fin X, x being set holds x in FinUnion(B,
  f) iff ex i being Element of X st i in B & x in f.i
proof
  let B be Element of Fin X, x be set;
A1: now
    assume x in union (f.:B);
    then consider Z being set such that
A2: x in Z and
A3: Z in f .:B by TARSKI:def 4;
    f.:B is Subset of Fin A by Lm2;
    then reconsider Z as Element of Fin A by A3;
    consider i being Element of X such that
A4: i in B & Z = f.i by A3,FUNCT_2:65;
    take i9 = i;
    thus i9 in B & x in f.i9 by A2,A4;
  end;
  now
    given i being Element of X such that
A5: i in B and
A6: x in f.i;
    f.i in f.:B by A5,FUNCT_2:35;
    hence x in union (f.:B) by A6,TARSKI:def 4;
  end;
  hence thesis by A1,Th46;
end;
