theorem Th46:
  for B being Element of Fin X holds FinUnion(B,f) = union (f.:B)
proof
  defpred X[Element of Fin X] means FinUnion($1,f) = union (f.:$1);
A1: for B being (Element of Fin X), i st X[B] holds X[B \/ {.i.}]
  proof
    let B be (Element of Fin X), i;
    assume FinUnion(B,f) = union (f.:B);
    hence FinUnion(B \/ {.i.}, f) = union (f.:B) \/ union {f.i} by Th45
      .= union (f.:B \/ {f.i}) by ZFMISC_1:78
      .= union (f.:B \/ Im(f,i)) by Th5
      .= union (f.:(B \/ {i})) by RELAT_1:120;
  end;
  FinUnion({}.X,f) = union (f.:{}.X) by Th44,ZFMISC_1:2;
  then
A2: X[{}.X];
  thus for B being Element of Fin X holds X[B] from FinSubInd3(A2,A1 );
end;
